Geometry of Moduli

The proceedings from the Abel Symposium on Geometry of Moduli, held at Svinøya Rorbuer, Svolvær in Lofoten, in August 2017, present both survey and research articles on the recent surge of developments in understanding moduli problems in algebraic geometry. Written by many of the main contributors to this evolving subject, the book provides a comprehensive collection of new methods and the various directions in which moduli theory is advancing. These include the geometry of moduli spaces, non-reductive geometric invariant theory, birational geometry, enumerative geometry, hyper-kähler geometry, syzygies of curves and Brill-Noether theory and stability conditions. Moduli theory is ubiquitous in algebraic geometry, and this is reflected in the list of moduli spaces addressed in this volume: sheaves on varieties, symmetric tensors, abelian differentials, (log) Calabi-Yau varieties, points on schemes, rational varieties, curves, abelian varieties and hyper-Kähler manifolds.


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Abel Symposia 14

Jan Arthur Christophersen Kristian Ranestad Editors

Geometry of Moduli

ABEL SYMPOSIA Edited by the Norwegian Mathematical Society

More information about this series at http://www.springer.com/series/7462

Jan Arthur Christophersen • Kristian Ranestad Editors

Geometry of Moduli

123

Editors Jan Arthur Christophersen Department of Mathematics University of Oslo Oslo, Norway

Kristian Ranestad Department of Mathematics University of Oslo Oslo, Norway

ISSN 2193-2808 ISSN 2197-8549 (electronic) Abel Symposia ISBN 978-3-319-94880-5 ISBN 978-3-319-94881-2 (eBook) https://doi.org/10.1007/978-3-319-94881-2 Library of Congress Control Number: 2018961007 Mathematics Subject Classification (2010): 14-02, 14D20, 14D22, 14C25, 14E99, 14H10, 14K10, 14L24, 18E30 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

The Norwegian government established the Abel Prize in mathematics in 2002, and the first prize was awarded in 2003. In addition to honoring the great Norwegian mathematician Niels Henrik Abel by awarding an international prize for outstanding scientific work in the field of mathematics, the prize shall contribute toward raising the status of mathematics in society and stimulate the interest for science among school children and students. In keeping with this objective, the Niels Henrik Abel Board has decided to finance annual Abel Symposia. The topic of the symposia may be selected broadly in the area of pure and applied mathematics. The symposia should be at the highest international level and serve to build bridges between the national and international research communities. The Norwegian Mathematical Society is responsible for the events. It has also been decided that the contributions from these symposia should be presented in a series of proceedings, and Springer Verlag has enthusiastically agreed to publish the series. The Niels Henrik Abel Board is confident that the series will be a valuable contribution to the mathematical literature. Chair of the Niels Henrik Abel Board

Kristian Ranestad

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Preface

The title of the Abel symposium 2017 was Geometry of Moduli and our goal was to highlight important recent developments in algebraic geometry regarding the theory of moduli. This included the geometry of moduli spaces, geometric invariant theory, birational geometry, enumerative geometry, hyper-Kähler geometry, and stability conditions. Moduli theory is ubiquitous in algebraic geometry, as can be seen in the list of moduli spaces treated in the lectures: sheaves on varieties, symmetric tensors, abelian differentials, (log) Calabi–Yau varieties, points on schemes, rational varieties, curves, abelian varieties, and hyper-Kähler manifolds. We believe the proceedings from the conference, which contain both original research and surveys of recent developments, reflect the breadth of and important recent advances in the field. The speakers and the titles of their lectures at the symposium were: • Arend Bayer: Bridgeland stability on Kuznetsov components in families • Jim Bryan: Donaldson-Thomas invariants of the banana manifold and elliptic genera • Ana-Maria Castravet: Derived categories of moduli spaces of stable rational curves • Dawei Chen: Geometry of moduli of abelian differentials • Izzet Coskun: The cohomology and birational geometry of moduli spaces of sheaves on surfaces • Barbara Fantechi: Infinitesimal deformations of log Calabi Yau varieties and orbifolds • Maksym Fedorchuk: Stability of Hilbert points and applications • Brendan Hassett: Rationality in families • Klaus Hulek: Degenerations of Hilbert schemes of degree 0 cycles on surfaces • Michael Kemeny: On the possible Betti tables of a canonical curve • Frances Kirwan: Applications of non-reductive geometric invariant theory • Emanuele Macri: Bridgeland stability and the genus of space curves • Kieran O’Grady: Abelian varieties associated to hyper-Kählers of Kummer type • Andrei Okounkov: Monodromy and derived equivalences vii

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Preface

• Aaron Pixton: Polynomiality of the double ramification cycle • Claire Voisin: Cubic fourfolds, hyper-Kähler manifolds and their degenerations The symposium took place from August 7 to 11, 2017, at Svinøya Rorbuer, Svolvær in Lofoten. The program and organizing committee consisted of Jan Arthur Christophersen (Oslo), John Christian Ottem (Oslo), Ragni Piene (Oslo), Kristian Ranestad (Oslo), Sofia Tirabassi (Bergen), Rahul Pandharipande (ETH Zurich), and Gavril Farkas (Humboldt, Berlin). We would like to express our gratitude to the Norwegian Mathematical Society for giving us the opportunity to host the Abel Symposium. We would also like to thank the administration of the Department of Mathematics, University of Oslo, for their assistance and Ruth Allewelt at Springer Verlag for her valued support in preparing these proceedings. Oslo, Norway Oslo, Norway May 8, 2018

Jan Arthur Christophersen Kristian Ranestad

Contents

Stratifying Quotient Stacks and Moduli Stacks . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Gergely Bérczi, Victoria Hoskins, and Frances Kirwan

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The Donaldson-Thomas Theory of K3 × E via the Topological Vertex . . . Jim Bryan

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An Extremal Effective Survey About Extremal Effective Cycles in Moduli Spaces of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Dawei Chen

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The Moduli Spaces of Sheaves on Surfaces, Pathologies and Brill-Noether Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Izzet Coskun and Jack Huizenga

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Geometric Invariant Theory of Syzygies, with Applications to Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 107 Maksym Fedorchuk The Topology of Ag and Its Compactifications . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 135 Klaus Hulek and Orsola Tommasi Syzygies of Curves Beyond Green’s Conjecture . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 195 Michael Kemeny GIT Versus Baily-Borel Compactification for Quartic K3 Surfaces . . . . . . . 217 Radu Laza and Kieran G. O’Grady Generalized Boundary Strata Classes . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 285 Aaron Pixton Torsion Points of Sections of Lagrangian Torus Fibrations and the Chow Ring of Hyper-Kähler Manifolds . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 295 Claire Voisin

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Stratifying Quotient Stacks and Moduli Stacks Gergely Bérczi, Victoria Hoskins, and Frances Kirwan

Abstract Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H ], where X is a projective scheme and H is a linear algebraic group with internally graded unipotent radical acting linearly on X, in such a way that each stratum [S/H ] has a geometric quotient S/H . This leads to stratifications of moduli stacks (for example, sheaves over a projective scheme) such that each stratum has a coarse moduli space.

1 Introduction Let H = U  R be a linear algebraic group over an algebraically closed field of characteristic 0 with internally graded unipotent radical U ; that is, the Levi subgroup R of H has a central one-parameter subgroup (1-PS) λ : Gm → R which acts on Lie U with all weights strictly positive. Of course any reductive group G has this form with both U and the central one-parameter subgroup λ being trivial; parabolic subgroups of reductive groups also have internally graded unipotent radicals in this sense, as do automorphism groups of complete toric varieties [2]. Suppose that H acts linearly on an irreducible projective scheme X with respect to an ample line bundle L over X. The aim of this paper is to describe a stratification of the quotient stack [X/H ] such that each stratum [S/H ] (where S is an H -invariant quasi-projective subscheme of X) has a geometric quotient S/H . When H = R

G. Bérczi Department of Mathematics, ETH Zürich, Zürich, Switzerland e-mail: [email protected] V. Hoskins Fachbereich Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany e-mail: [email protected] F. Kirwan () Mathematical Institute, Oxford University, Oxford, UK e-mail: [email protected] © Springer Nature Switzerland AG 2018 J. A. Christophersen, K. Ranestad (eds.), Geometry of Moduli, Abel Symposia 14, https://doi.org/10.1007/978-3-319-94881-2_1

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is reductive this stratification refines the Hesselink–Kempf–Kirwan–Ness (HKKN) stratification associated to the linear action on X (cf. [14, 18, 19, 23]). Potential applications of this construction include moduli stacks which can be filtered by quotient stacks with compatible linearisations; for example, it can be applied to moduli of sheaves of fixed Harder–Narasimhan type over a projective scheme [7], and moduli of unstable projective curves [17]. When H is reductive Mumford’s geometric invariant theory (GIT) [22] allows us to find open subschemes Xs ⊆ Xss of X, the stable and semistable loci, such that Xs has a geometric quotient Xs /H and Xss has a good quotient ⎛ φ : Xss → X//H = Proj ⎝



⎞ H 0 (X, L⊗m )H ⎠ .

m0

Here Xs = Xs,H = Xs,H,L and Xss = Xss,H = Xss,H,L depend on the choice of linearisation L (that is, the ample line bundle L and the lift of the action of H to an action on L) and the GIT quotient X//H = X//L H is a projective scheme with Xs /H as an open subscheme. Moreover when x, y ∈ Xss then φ(x) = φ(y) if and only if the closures of the H -orbits of x and y meet in Xss . The Hilbert–Mumford criteria allow us to determine the stable and semistable loci in a simple way without  needing to understand the algebra of invariants m0 H 0 (X, L⊗m )H . The best situation occurs when Xss = Xs = ∅; then X//H = Xs /H is both a projective scheme and a geometric quotient of the open subscheme Xs of X. More generally if Xs = ∅ then the projective completion X//H of the geometric quotient  ss /H where ψ : X ss → Xss is Xs /H has a ‘partial desingularisation’ X//H =X ss s ss obtained as follows [20]. If X = X then there exists x ∈ X whose stabiliser in ss we first blow up H is reductive of dimension strictly bigger than 0. To construct X ss X along its closed subscheme where the stabiliser has maximal dimension in Xss (such stabilisers are always reductive) or equivalently blow up X along the closure of this subscheme in X. We then remove the complement of the semistable locus for a small ample perturbation of the pullback linearisation. The maximal dimension of a stabiliser in this new semistable locus is strictly less than in Xss . When Xs = ∅, ss = X s = ∅ and the partial repeating this process finitely many times leads to X s   /H . desingularisation X//H =X  When H is non-reductive, then the graded algebra m0 H 0 (X, L⊗m )H is not necessarily finitely generated and in general the attractive properties of Mumford’s GIT fail [3]. However when the unipotent radical U of H = U  R is graded in the sense described above by a central 1-PS λ : Gm → R of the Levi subgroup R, then after twisting the linearisation by an appropriate rational character, so that it becomes ‘graded’ itself in the sense of [5], some of the desirable properties of classical GIT still hold [4, 6]. More precisely, we first quotient by the linear action := U  λ(Gm ) using the results of [4, 6] described of the graded unipotent group U ∼ in Sect. 2.3, then we quotient by the residual action of the reductive group H /U = R/λ(Gm ). In the best case, when the U -action is free on a certain open subscheme

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0 Xmin of X (cf. the condition (∗) in Definition 5 and Theorem 2.7), one can construct -action a geometric quotient of an open subscheme of ‘stable points’ for the U such that the quotient is projective and this stable set has a Hilbert–Mumford type description. If the U -action has positive dimensional stabilisers generically, one can conclude the same results if we assume a weaker condition (∗ ∗ ∗) (cf. Theorem 2.11). Even when this weaker condition fails, one can perform an iterated →X sequence of blow-ups of X along H -invariant subschemes to obtain ψ : X  has an induced linear H -action satisfying (∗ ∗ ∗). Hence, there is a such that X -quotient of an open subscheme of X  that contains as projective and geometric U -quotient of an open subscheme of X, as ψ is an an open subscheme a geometric U isomorphism away from the exceptional divisor. Now suppose that G is a reductive group acting linearly on a projective scheme X with respect to an ample line bundle L. Associated to this linear G-action and an invariant inner product on Lie G, there is a stratification (the ‘HKKN stratification’ cf. [14, 18, 19, 23])

X=





β∈B

of X by locally closed subschemes Sβ , indexed by a partially ordered finite set B, such that such that S0 = Xss , 1. if Xss = ∅, then B has a minimal element 0 2. for β ∈ B, the closure of Sβ is contained in β β Sβ , and 3. for β ∈ B, we have Sβ ∼ = G×Pβ Yβss , where G×Pβ Yβss is the quotient of G×Yβss by the diagonal action of a parabolic subgroup Pβ of G acting on the right on G and on the left on a Pβ -invariant locally closed subscheme Yβss of X. In fact, Yβss is an open subscheme of a projective subscheme Y β of X that is determined by the action of a Levi subgroup Lβ of Pβ with respect to the restriction of the G-linearisation L → X to the Pβ -action on Y β twisted by a rational character χβ of Pβ . The index β determines a central (rational) 1-PS λβ : Gm → Lβ and χβ is the corresponding rational character, where the choice of invariant inner product allows us to identify characters and co-characters of a fixed maximal torus (cf. Remark 2.2). Furthermore Pβ is the parabolic subgroup P (λβ ) determined by the 1PS λβ , which grades the unipotent radical Uβ of Pβ . Thus by Property (3) above, to construct a G-quotient of (an open subset of) an unstable stratum Sβ , we can study the linear Pβ -action on Y β and apply the results described above for the action of := Uβ  λβ (Gm ). U The G-action on the stratum Sβ has a categorical quotient Zβ //Lβ induced by the morphism ss,Lβ /λβ (Gm )

pβ : Yβss → Zβss = Zβ

y → pβ (y) := lim λβ (t)y, t →0

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where Zβ is the union of those connected components of the fixed point set for the action of λβ (Gm ) on X over which λβ acts on the fibres of L with weight given by the restriction of β. However this categorical quotient is in general far from being a geometric quotient; it identifies y with pβ (y), which lies in the orbit closure but typically not the orbit of y. In this article we will show that, applying the blow-up sequence needed to construct a quotient by an action of a linear algebraic group with internally graded unipotent radical to the Pβ -action on the projective subscheme Y β of X, we can refine the stratification {Sβ |β ∈ B} to obtain a stratification of X such that each stratum is a G-invariant quasi-projective subscheme of X with a geometric quotient by the action of G. This refined stratification is a further refinement of the construction described in [21]. The quotient stack [X/G] has an induced stratification {Σγ |γ ∈ Γ } such that each stratum Σγ has the form Σγ ∼ = [Wγ /Hγ ] where Wγ is a quasi-projective subscheme of X acted on by a linear algebraic subgroup Hγ of G with internally graded unipotent radical, and this action has a geometric quotient Wγ /Hγ . Moreover under appropriate hypotheses (involving condition (∗ ∗ ∗)) for the actions of the subgroups Hγ , the geometric quotients Wγ /Hγ are themselves projective. This will follow from the following theorem, which is proved in Sect. 3. Theorem 1.1 Let H = U  R be a linear algebraic group with internally graded unipotent radical U acting on a projective scheme X over an algebraically closed field k of characteristic 0 with respect to an ample linearisation and fix an invariant inner product on Lie R. Then X has a stratification {Sγ |γ ∈ Γ } induced by the linearisation L and grading λ : Gm → R for the action of H on X, such that the following properties hold. (i) The index set Γ is finite and partially ordered such that for all γ ∈ Γ , we have Sγ ⊆ Sγ ∪



Sδ .

δ∈Γ,δ>γ

(ii) Each Sγ is an H -invariant quasi-projective subscheme of X with a geometric quotient Sγ /H . (iii) If Y is an H -invariant projective subscheme of X then the stratification {SγY |γ ∈ Γ Y } of Y induced by the restriction L |Y of the linearisation L to Y is (up to taking connected components) the restriction to Y of the stratification {Sγ |γ ∈ Γ } of X, so that there is a map of indexing sets φY : Γ Y → Γ such that if γ ∈ Γ Y then SγY is a connected component of SφY (γ ) ∩ Y ; Moreover, if H = G is reductive, then this stratification satisfies the following additional properties.

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(iv) The stratification {Sγ |γ ∈ Γ } is a refinement of the HKKN stratification {Sβ |β ∈ B} for the linearisation L (cf. Sect. 2.1.1). (v) If β ∈ B (which we recall determines a 1-PS λβ : Gm → Pβ ≤ G) satisfies x ∈ Zβss



dim(StabG (x)/λβ (Gm )) = 0,

(†)

then the connected components of GZβss and Sβ \ GZβss (if these are nonempty) are strata in the refined stratification {Sγ |γ ∈ Γ }. As a consequence, we obtain a stratification of the quotient stack [X/H ] by locally closed substacks, each of which admits a coarse moduli space (cf. Corollary 1). Inspired by the reductive GIT notion of a good quotient, Alper introduces a notion of a good moduli space for a stack [1]. However, in general the strata appearing in this stratification of [X/H ] will not admit good moduli spaces, because a necessary condition for a stack to admit a good moduli space is that its closed points have reductive stabiliser groups (cf. [1, Proposition 12.14]). In general (even when H = G is reductive) the points in the strata of [X/H ] will have nonreductive stabiliser groups. If H = G is reductive, then a stacky version of the HKKN stratification has been studied by Halpern-Leister, and by abstracting this concept he obtains a notion of a Θ-stratification [12]. Indeed, the linearisation of the G-action on X and the choice of invariant inner product is precisely the data required to construct a Θ-stratification of [X/G], and this Θ-stratification is the stratification {[Sβ /G] : β ∈ B} obtained from the HKKN stratification of X. The stratification described above thus refines this Θ-stratification without depending on any additional data. Since the construction of the refined stratification involves studying the blowup procedures used in partial desingularisations of reductive GIT quotients [20] and for constructing geometric quotients by linear algebraic groups with internally graded unipotent radical [6], one can ask how this compares with the stack-theoretic blow-up constructions. The ideas in [20] have been generalised to stacks by Edidin and Rydh [11] to show that for a smooth Artin stack X admitting a stable good moduli space, there is a sequence of birational morphisms of smooth Artin stacks Xn → · · · X1 → X such that the good moduli space of Xn is an algebraic space with only tame quotient singularities and is a partial desingularisation of the good moduli space of X. However, for H non-reductive, it is often the case that [X/H ] will not have a good moduli space, and so one cannot apply this result. The picture provided by Theorem 1.1 has potential applications to moduli stacks which are filtered by quotient stacks, and to the construction of moduli spaces of ‘unstable’ objects (for example, moduli of sheaves over a projective scheme [7] or moduli of projective curves [17]). Suppose that M is a moduli stack which can be expressed as an increasing union M =

n0

Un

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of open substacks of the form Un ∼ = [Vn /Gn ] where [Vn /Gn ] is the quotient stack associated to a linear action on a quasiprojective scheme Vn by a group Gn which is reductive (or more generally has internally graded unipotent radical). We can look for suitable ‘stability conditions’ on M : linearisations (Ln )n0 for the actions of Gn on projective completions Vn of Vn and invariant inner products on Lie Gn which are compatible in the sense that the stratification induced by Ln on [Vn /Gn ] restricts to the stratification induced by Lm on [Vm /Gm ] when n > m. This situation arises for sheaves over a projective scheme [7, 16], for example, and also for projective curves [17], and we obtain a stratification {Σγ |γ ∈ Γ } of the stack M such that each stratum Σγ is isomorphic to a quotient stack [Wγ /Hγ ], where Wγ is quasi-projective acted on by a linear algebraic group Hγ with internally graded unipotent radical, and there is a geometric quotient Wγ /Hγ which is a coarse moduli space for Σγ . The geometric quotient Wγ /Hγ will be projective if semistability coincides with stability in an appropriate sense for the action of Hγ on a suitable projective completion of Wγ with respect to an induced linearisation. The layout of this article is as follows. In Sect. 2, we will review classical and non-reductive GIT, describing how to construct quotients by actions of linear algebraic groups with internally graded unipotent radical. The heart of the paper is Sect. 3, in which we describe how to stratify a quotient stack [X/H ] into strata Σγ = [Wγ /Hγ ] where the action of Hγ on Wγ has a geometric quotient Wγ /Hγ . The argument is an inductive one, so the assumption on X and H is that H is a linear algebraic group with internally graded unipotent radical, and X is a projective scheme which has an amply linearised action of H . In Sect. 4, this construction is applied to stacks which are suitably filtered by quotient stacks, and Sect. 5 contains a brief discussion of examples including moduli of unstable curves and moduli of sheaves of given Harder–Narasimhan type over a fixed projective scheme. We would like to thank Brent Doran, Daniel Halpern-Leistner, Eloise Hamilton and Joshua Jackson for helpful discussions about this material.

2 Classical and Non-reductive GIT 2.1 Classical GIT for Reductive Groups In Mumford’s GIT [22], a linearisation of an action of a reductive group G on a projective scheme X over an algebraically closed field k of characteristic 0 is given by a line bundle L (which we will always assume to be ample) on X and a lift of the action to L. Since G is reductive, the algebra of G-invariant sections L (X)G is finitely generated as a graded algebra with associated projective scheme O

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L (X)G ). X//G = Proj(O  L (X) := ∞ H 0 (X, L⊗k ) (X, L)  O k=0 | | | ↓ L (X)G algebra of invariants. X//G  O L (X)G in O L (X) determines a rational map X − − → X//G The inclusion of O which fits into a diagram X −− → X//G projective || semistable Xss −−−− X//G open stable Xs −−−− Xs /G where Xs (the stable locus) and Xss (the semistable locus) are open subschemes of X, there is a geometric quotient Xs /G for the action of G on Xs , the GIT quotient X//G is a good quotient for the action of G on Xss via the G-invariant surjective morphism φ : Xss → X//G, and φ(x) = φ(y) ⇔ Gx ∩ Gy ∩ Xss = ∅. The semistable and stable loci Xss and Xs of X are characterised by the following properties ([22, Chapter 2], [24]). Proposition 2.1 (Hilbert–Mumford Criteria for Reductive Groups) Let T be a maximal torus of G. 1. A point x ∈ X is semistable (respectively stable) for the G-action on X if and only if for every g ∈ G the point gx is semistable (respectively stable) for the T -action. 2. A point x ∈ X ⊂ Pn with homogeneous coordinates [x0 : . . . : xn ] is semistable (respectively stable) for a diagonal T -action on Pn with weights α0 , . . . , αn if and only if 0 ∈ Conv{αi : xi = 0} (respectively 0 is contained in the interior of this convex hull).

2.1.1 The HKKN Stratification Associated to the linear action of G on X and an invariant inner product on the Lie algebra of G, there is a stratification (the ‘HKKN stratification’, which in the case

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k = C is the Morse stratification for the norm-square of an associated moment map [14, 18, 19, 23]). Remark 2.2 Let us clarify what is meant by this invariant inner product, whose associated norm we denote by || − ||. If k = C, then G is the complexification of its maximal compact group K; then the Lie algebra of K is a real vector space, and we choose an inner product on this Lie algebra that is invariant under the adjoint action of K. In fact, we will also assume that we fix a maximal compact torus Tc ⊂ K such that the inner product is integral on the co-character lattice X∗ (Tc ) ⊂ Lie K. For an arbitrary algebraically closed field k of characteristic zero, one can fix a maximal torus T of G and choose an inner product on the co-character space X∗ (T ) ⊗Z R that is invariant for the Weyl group of T and is integral on the co-character lattice (for example, see [15, §2]). Then this inner product gives an identification between characters and co-characters (i.e. 1-PSs) of T . The HKKN stratification associated to the action of G on X with respect to L and the norm || − || is a stratification X=





β∈B

of X by locally closed subschemes Sβ , indexed by a partially ordered finite subset B of rational elements in a positive Weyl chamber for the reductive group G, with the following properties. 1. If 0 ∈ B, then this is the minimal element and S0 = Xss . Moreover, for each β ∈ B, we additionally have the following properties: 2. the closure of Sβ is contained in β β Sβ where γ  β if and only if γ = β or ||γ || > ||β||; 3. Sβ ∼ = G ×Pβ Yβss := (G × Yβss )/Pβ where this quotient is of the diagonal action of Pβ on the right on G and on the left on Yβss . Here Pβ is a parabolic subgroup of G which acts on a locally closed subscheme Yβss of X. More precisely, β ∈ B determines a (rational) 1-PS λβ : Gm → G and an associated parabolic subgroup Pβ = P (λβ ) = Uβ  Lβ with Levi subgroup Lβ = StabG (β) such that the conjugation action of λβ (Gm ) on Lie Uβ has strictly positive weights; thus Pβ has internally graded unipotent radical. Let Zβ be the union of components in the fixed locus Xλβ (Gm ) on which this 1-PS acts on the fibres of L with weight given by the restriction of β, and let Yβ ⊂ X be the subscheme of points x ∈ X such that limt →0 λβ (t)y ∈ Zβ ; thus there is a retraction pβ : Yβ → Zβ

y → pβ (y) := lim λβ (t)y t →0

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which is equivariant with respect to the quotient homomorphism qβ : Pβ → Lβ obtained by identifying Lβ with Pβ /Uβ . Let Lβ denote the restriction of the Glinearisation L on X to the Pβ -action on Y β twisted by the (rational) character χβ of Pβ corresponding to the 1-PS λβ (via the norm || − ||). We also let Lβ denote the restriction of this linearisation to the Lβ -action on Zβ . Then Yβss (respectively Zβss ) is the semistable locus for the Pβ -action on Y β (respectively the Lβ /λβ (Gm )-action on Zβ ) linearised by the twisted linearisation Lβ ; furthermore, Yβss = pβ−1 (Zβss ). Finally, we make the following observation about quotienting the unstable strata (cf. [16]). Remark 2.3 The G-action on Sβ has a categorical quotient Zβ //Lβ induced by the map pβ : Yβss → Zβss . In general, this quotient is far from being a geometric quotient (even after restriction to the pre-image of any nonempty open subscheme of Zβ //Lβ ), as y ∈ Yβss is identified with pβ (y) ∈ Gy. By (3), constructing a quotient of the G-action on a G-invariant open subset of Sβ is equivalent to constructing a Pβ -quotient of a Pβ -invariant open subset of Yβss (or its closure); the latter perspective will lead to a geometric quotient by using GIT for the non-reductive group Pβ , whose unipotent radical Uβ is internally graded by λβ (cf. Sect. 2.3).

2.1.2 Partial Desingularisations of Reductive GIT Quotients The geometric quotient Xs /G has at most orbifold singularities when X is nonsingular, since the stabiliser subgroups of stable points are finite subgroups of G. If Xss = Xs = ∅, the singularities of X//G are typically more severe even when  X is itself nonsingular, but X//G has a ‘partial desingularisation’ X//G which is s also a projective completion of X /G and is itself a geometric quotient  ss /G X//G =X s of a G-equivariant blow-up X  of X [20]. ss = X by G of an open subscheme X ss ss  Here X is obtained from X by successively blowing up along the subschemes of semistable points stabilised by reductive subgroups of G of maximal dimension and removing the complement of the semistable locus from the resulting blow-up.  For the construction of the partial desingularisation X//G in [20], it is assumed s that X = ∅. There exist semistable points of X which are not stable if and only if there exists a non-trivial connected reductive subgroup of G fixing a semistable point. Let r > 0 be the maximal dimension of a reductive subgroup of G fixing a point of Xss and let R(r) be a set of representatives of conjugacy classes of all connected reductive subgroups R of dimension r in G such that ZRss := {x ∈ Xss : Rx = x}

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is non-empty. Then ss ZR (r) :=



GZRss

R∈R (r)

is a disjoint union of closed G-invariant subschemes of Xss . The action of G on ss Xss lifts to an action on the blow-up X(1) of Xss along ZR (r) , and this action can ss be linearised so that the complement of X(1) in X(1) is the proper transform of the closed subscheme π −1 (π(GZRss )) of Xss where π : Xss → X//G is the quotient map (see [20, 7.17]). The G-linearisation on X(1) used here is (a tensor power of) the pullback of the ample line bundle L on X along ψ(1) : X(1) → X perturbed by a sufficiently small multiple of the exceptional divisor E(1) ; then, if the perturbation is sufficiently small, we have −1 −1 s ss (Xs ) ⊆ X(1) ⊆ X(1) ⊆ ψ(1) (Xss ) = X(1) , ψ(1) s and X ss will be independent of the choice and the stable and semistable loci X(1) (1) ss is fixed by a reductive subgroup of of perturbation. Moreover, no point x ∈ X(1) ss is fixed by a reductive subgroup R of G of dimension at least r, and x ∈ X(1) dimension less than r in G if and only if it belongs to the proper transform of the closed subscheme ZRss of Xss . ss ss of ZR in X (or Remark 2.4 In [20], X itself is blown up along the closure ZR (r) (r) ss in a projective completion of X with a G-equivariant morphism to X which is an isomorphism over Xss ). This gives a projective scheme X (1) and blow-down map ψ(1) : X(1) → X restricting to ψ(1) : X(1) → X where (ψ(1) )−1 (Xss ) = X(1) . For a sufficiently small perturbation of the pullback to X (1) of the linearisation on s ss X, we have (ψ(1) )−1 (Xs ) ⊆ X(1) ⊆ X (1) ⊆ (ψ(1) )−1 (Xss ) = X(1) , and moreover the restriction of the linearisation to X(1) is obtained from the pullback of L by perturbing by a sufficiently small multiple of the exceptional divisor E(1) . ss ss to obtain X(2) such that no If r > 1, we can apply the same procedure to X(1) ss point of X(2) is fixed by a reductive subgroup of G of dimension at least r − 1. If ss → Xss such Xs = ∅ then repeating this process at most r times gives us ψ : X s that ψ is an isomorphism over X and no positive-dimensional reductive subgroup ss . The partial desingularisation X//G  ss /G can be of G fixes a point of X = X ss obtained by blowing up X//G along the proper transforms of π(GZR ) ⊂ X//G in decreasing order of the dimension of R.

Remark 2.5 Suppose for simplicity that X is irreducible. a) If Xs = ∅, then this is the situation considered in [20] and the partial  = X. desingularisation construction is described above. If Xss = Xs , then X ss b) If X = ∅, then there is an unstable stratum Sβ with β = 0 in the HKKN stratification (cf. Sect. 2.1.1) which is a non-empty open subscheme of X, and thus when X is irreducible X = Sβ . Then constructing a quotient of a

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non-empty open subscheme of X reduces to non-reductive GIT for the action of the parabolic subgroup Pβ on Yβ as described in Sect. 2.3 below, where a blow-up sequence may also need to be performed. c) If Xs = ∅ = Xss then the partial desingularisation construction can be applied to Xss , and there are different ways in which it can terminate. (i) If Xss = GZRss ∼ = G×NR ZRss for a positive-dimensional connected reductive subgroup R of G with normaliser NR in G, then NR and NR /R are also reductive, and X//G ∼ = ZR //(NR /R) = ZR //NR ∼ where ZR is the closed subscheme of X which is the fixed point set for the action of R. Then we can apply induction on the dimension of G to study this case. (ii) If GZRss = Xss for each positive-dimensional connected reductive subgroup R of G, then we can perform the first blow-up in the partial desingularisation ss ⊆ X construction to obtain ψ(1) : X(1) → Xss such that X(1) (1) and s s s ∼ X(1) = ∅ as above (as X(1) is open and X(1) \ E(1) = Xs = ∅, where ss = ∅, then X E(1) is the exceptional divisor). If X(1) (1) has a dense open ss = GZ ss stratum S(1),β for β = 0 as in Case b). If we have X(1) (1),R for a positive-dimensional connected reductive subgroup R of G, where ss ss : Rx = x}, then we proceed as in Case (i) above. Z(1),R = {x ∈ X(1) Otherwise we can repeat the process, until it terminates in one of these two ways.

2.2 GIT for Non-reductive Groups Now suppose that X is a projective scheme over an algebraically closed field k of characteristic 0 and let H be a linear algebraic group, with unipotent radical U , acting on X with respect to an ample linearisation L. Definition 1 (Semistability for the Unipotent Group cf. [10, §4] and [10, 5.3.7]) For an invariant section f ∈ I = m>0 H 0 (X, L⊗m )U , let Xf be the U -invariant affine open subset of X on which f does not vanish, and let O(Xf ) denote its coordinate ring. ss,U = 1. The semistable locus for the U -action on X linearised by L is X f ∈I fg Xf where

I fg = {f ∈ I | O(Xf )U is finitely generated}. lts,U = 2. The (locally trivial) stable locus for the linearised U -action on X is X f ∈I lts Xf where

I lts := {f ∈ I fg | qU : Xf → Spec(O (Xf )U ) is a locally trivial geometric quotient}.

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3. The enveloped quotient of Xss,U by the linear U -action is qU : Xss,U → L (X)U ) is the natural morphism of qU (Xss,U ), where qU : Xss,U → Proj(O ss,U schemes and qU (X ) is a dense constructible subset of the enveloping quotient X U= Spec(O(Xf )U ). f ∈I f g

Remark 2.6









L (X)U is finitely generated, so that X U = Proj(O L (X)U ), the 1. Even when O ss,U enveloped quotient qU (X ) is not necessarily a subscheme of X U (for example, see [10, §6]). 2. The enveloping quotient X U has quasi-projective open subschemes (‘inner enveloping quotients’ X//◦ U ) that contain the enveloped quotient qU (Xss ) and have ample line bundles which under the natural map qU : Xss → X U pull back to positive tensor powers of L (see [3] for details). The H -semistable locus Xss = Xss,H and enveloped and (inner) enveloping quotients H



qH : Xss → qH (Xss ) ⊆ X//◦ H ⊆ X

for the linear action of H are defined as for the unipotent case in Definition 1 and Remark 2.6 (cf. [3]), but the definition of the stable locus Xlts = Xlts,H for the linear action of H combines (and extends) the definitions for unipotent and reductive groups. Definition 2 (Stability for Linear Algebraic Groups) Let H be a linear algebraic group acting linearly on X with respect to an ample line bundle L; then the (locally trivial) stable locus is the open subscheme Xlts = f ∈I lts Xf of Xss , where I lts ⊆ 0 ⊗r H r>0 H (X, L ) is the subset of H -invariant sections f satisfying the following conditions: 1. the H -invariant open subscheme Xf is affine; 2. the H -action on Xf is closed with finite stabiliser groups; and 3. the restriction of the U -enveloping quotient map L (X)U )(f ) ) qU : Xf → Spec((O is a locally trivial geometric quotient for the U -action on Xf .

2.3 GIT for Linear Algebraic Groups with Internally Graded Unipotent Radicals Now suppose that H = U  R is a linear algebraic group with internally graded unipotent radical U in the following sense.

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Definition 3 We say H = U  R has internally graded unipotent radical U if there is a central 1-PS λ : Gm → R whose conjugation action on the Lie algebra of U = U  λ(Gm )  H for the associated has strictly positive weights. We write U semi-direct product. Suppose also that H acts linearly on X with respect to an ample line bundle L. It is shown in [4, 6] that the algebra of H -invariant sections is finitely generated provided: (a) L is replaced with a suitable tensor power L⊗m , with m ≥ 1 sufficiently divisible, and the linearisation of the action of H is twisted by a suitable (rational) character, and (b) condition (∗) described below (also known as ‘semistability coincides with stability for the unipotent radical U ’) holds.

≈≈

Moreover, in this situation the natural quotient morphism qH from the semistable locus Xss,H to the enveloping quotient X H is surjective, and expresses the projective scheme X//H = X//◦ H = X H as a good quotient of Xss,H . Furthermore this locus Xss,H can be described using Hilbert–Mumford criteria. It is also shown in [6] that when condition (∗) is not satisfied, but is replaced with a slightly weaker condition, such as (∗∗) below, then there is a sequence of blow-ups of X along H -invariant subschemes (similar to that of [20] when H is reductive) with an induced linear action of H satisfying resulting in a projective scheme X condition (∗). In fact, these results can be generalised to allow actions where the U -action has positive dimensional stabilisers (cf. Theorems 2.11 and 2.12 below). Before giving a precise description of the condition (∗) and its variants, we define the notion of an adapted linearisation, which is also needed for the statement of the main results of [4, 6]. of χ contains U Let χ : H → Gm be a character of H ; the restriction to U in its kernel and can be identified naturally with an integer so that the integer 1 which fits into the exact sequence U → U  corresponds to the character of U λ(Gm ). By replacing L with a sufficiently high power, we can without loss of generality assume that L is very ample. Let ωmin := ω0 < ω1 < · · · < ωmax be the ≤ H acts on the fibres of the tautological weights with which the 1-PS λ : Gm → U line bundle OP((H 0 (X,L)∗) (−1) over points of the fixed locus P((H 0 (X, L)∗ )λ(Gm ) . Without loss of generality we may assume that there exist at least two distinct such weights, as otherwise the grading hypothesis implies that the U -action on X is trivial, in which case we can take a quotient by the action of the reductive group R = H /U . Definition 4 For a character χ of H as above and a positive integer c, we say the if rational character χ/c is adapted to the linear action of U ωmin := ω0 <

χ < ω1 . c

-action if ωmin := ω0 < 0 < ω1 . Furthermore, we say L is adapted to the U

(1)

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, then the If the rational character χ/c is adapted to the linear action of U H -linearisation Lχ⊗c on X given by twisting the ample line bundle L⊗c by the character χ (that is, so that the weights ωj are replaced with ωj c − χ) is adapted. s,λ(Gm ) s,Gm Let Xmin denote the stable locus in X for the linear action of Gm + = Xmin + via λ with respect to the adapted linearisation Lχ⊗c and, for a maximal torus T of (s)s,T H containing λ(Gm ), let Xmin + denote the (semi)stable locus in X for the linear action of T with respect to the adapted linearisation L⊗c χ . By the theory of variation s,λ(Gm ) ss,λ(Gm ) = Xmin is independent of (classical) GIT [9, 28], the stable locus Xmin + + of the choice of adapted rational character χ/c. In fact, by the Hilbert–Mumford s,λ(G ) 0 criterion, we have Xmin +m = Xmin \ Zmin , where if Vmin denotes the weight space 0 ∗ of ωmin in V = H (X, L) , then 

Zmin := X ∩ P(Vmin ) = x ∈ Xλ(Gm ) : λ(Gm ) acts on L∗ |x with weight ωmin

and 0 Xmin := {x ∈ X |

lim

t →0, t ∈Gm

λ(t) · x ∈ Zmin }.

Definition 5 (Conditions (∗)–(∗ ∗ ∗) Generalising ‘Semistability Equals Stability’ cf. [6]) With the above notation, we define the following conditions for the -action on X. U StabU (z) = {e} for every z ∈ Zmin . 0 StabU (x) = {e} for generic x ∈ Xmin .

(∗) (∗∗)

Moreover, if U  U (1)  . . .  U (s)  {e} denotes the derived series of U , we define condition 0 . for 1 ≤ j ≤ s, there exists dj ∈ N such that dim StabU (j) (x) = dj for all x ∈ Xmin

(∗ ∗ ∗)

0 . Note that condition (∗) holds if and only if we have StabU (x) = {e} for all x ∈ Xmin This condition is also referred to in [6] as the condition ‘semistability coincides with (or, when the 1-PS λ : Gm → R is fixed, for the linear stability’ for the action of U action of U ).

Definition 6 Let T ≤ R be a maximal torus containing λ(Gm ). The min-stable (respectively min-semistable) locus for a linear H -action satisfying condition (∗ ∗ ∗) is for the action of the graded unipotent group U  s,H s,T hXmin Xmin + := + h∈H ss,H (respectively Xmin + :=

 h∈H

ss,T hXmin + ).

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-action then satisfies The min-stable locus for the U  s,λ(G ) s,U ss,U 0 Xmin uXmin +m = Xmin \ U Zmin . + = Xmin + = u∈U

) Theorem 2.7 (U-Theorem When Semistability Coincides with Stability for U [6] Suppose that the linearisation for the action of H on X is adapted as in -action on X satisfies condition (∗). Then the following Definition 4 and that the U statements hold. s,U (i) The open subscheme Xmin + of X has a projective geometric quotient X//U = s,U by U . /U X min +

ss,H (ii) The open subscheme Xmin + of X has a good quotient X//H (X//U )//(R/Gm ) by H = U  R, which is also projective.

=

Remark 2.8 In the proof of Theorem 2.7 (and its variants below), one replaces the adapted linearisation by a ‘well adapted’ linearisation (which can be achieved by twisting by a rational character); this is a slightly stronger notion. This strengthening s,U does not alter Xmin + or its quotient X//U , but it affects what can be said about and X//H = (X//U )//(R/λ(Gm )). The induced ample line bundles on X//U U proofs in [4, 6] that the algebras of invariants ⊕m≥0 H 0 (X, L⊗cm mχ ) and 

H H 0 (X, L⊗cm mχ ) = (

m≥0





U (R/λ(Gm )) H 0 (X, L⊗cm mχ ) )

m≥0

= Xs,U /U and are finitely generated, and that the enveloping quotients X//U min + X//H are the associated projective schemes, require that the linearisation is twisted by a well adapted rational character χ/c. More precisely, it is shown in [4, 6] that, given a linear action of H on X with respect to an ample line bundle L, there exists  > 0 such that if χ/c is a rational character of Gm (lifting to H ) with c sufficiently divisible and χ : H → Gm a character of H such that ωmin <

χ < ωmin + , c

U 0 ⊗cm H then the algebras of invariants ⊕m≥0 H 0 (X, L⊗cm mχ ) and ⊕m≥0 H (X, Lmχ ) are and X//H satisfy finitely generated, and the associated projective schemes X//U the conclusions of Theorem 2.7.

Theorem 2.7 describes the good case when semistability coincides with stability . The following versions proved in [6] apply more generally. for the linear action of U Theorem 2.9 (U-Theorem Giving Projective Completions [6]) Suppose that the on X is adapted and satisfies condition (∗∗). Then there exists linear action of U a sequence of blow-ups along H -invariant projective subschemes, resulting in a

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(with blow-down map ψ → X) such that the conditions of :X projective scheme X Theorem 2.7 hold for a suitable ample linearisation. U )s,R/λ(Gm) for the induced linear action of If in addition the stable locus (X// U is nonempty, then this sequence of blow-ups along H -invariant R/λ(Gm ) on X//  projective subschemes of X can be extended, resulting in a projective scheme X  → X) such that the conditions of Theorem 2.7 still  :X (with blow-down map ψ hold for a suitable ample linearisation, and such that the quotient given by that s,H of X.  theorem is a geometric quotient of an open subscheme X Remark 2.10 For the first stage of this blow-up sequence the centres are determined to obtain X, while for the second stage by considering the stabiliser subgroups for U one blows up by considering the stabiliser subgroups for the reductive group R to  obtain X. → X such :X In the first step, one performs a blow-up sequence to obtain ψ that the U -action on X satisfies condition (∗) with respect to a linearisation L , ∗ (L ). The centres of the blow-ups which is an arbitrarily small perturbation of ψ of used to obtain X from X are determined by the dimensions of the stabilisers in U 0 x ∈ Xmin (for details, see [6]). Then one can construct a projective and geometric -action on X s,U by Theorem 2.7 and, as ψ is an isomorphism quotient of the U min + -invariant away from the exceptional divisor, one obtains a geometric quotient of a U sˆ ,U s ˆ , U U , where X of X as an open subscheme of X// is the image open subset X min +

min +

s,U with the complement of the exceptional divisor of the intersection of X under ψ min + Another characterisation of XMs,U , when StabU (z) = {e} for generic z ∈ in X. Zmin , is as s,U ) | dim StabU (lim λ(t) · x) = 0}; {x ∈ ψ(X min + t →0

= X and Xsˆ ,U = Xs,U . If one is only interested in obtaining if (∗) holds then X min + min + a good quotient for the H -action, then the second stage of the blow-up procedure is not needed: one can then take a reductive GIT quotient of the residual action on U of H /U = R/λ(Gm ), and thus one obtains a good quotient of the H -action X// L U )//(R/λ(Gm )). Moreover, on an open subset of X as an open subscheme of (X// this good quotient restricts to a geometric quotient on an open subscheme of stable points. to the blow-up X  in Theorem 2.9, one performs an additional blow To go from X up sequence by considering the stabiliser groups for the action of the reductive group R/λ(Gm ) as in the partial desingularisation procedure described in Sect. 2.1.2. This gives us an H -invariant open subscheme of X with a geometric quotient by H which  H (and also of is isomorphic to an open subscheme of the projective scheme X// L X//LH ). Theorem 2.7 can be generalised by weakening the condition (∗) further to (∗ ∗ ∗) to allow for actions with positive dimensional stabiliser groups generically.

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-Theorem with Positive-Dimensional Stabilisers in U ) [6] Theorem 2.11 (U -action on X. Then Suppose that condition (∗ ∗ ∗) holds for an adapted linear U the conclusions of Theorem 2.7 hold. In fact, this theorem still holds if we replace the derived series in condition (∗ ∗ ∗) with any series U > U (1) > . . . > U (s) > {e} which is normalised by H and whose successive quotients U (j ) /U (j +1) are abelian, provided that (∗ ∗ ∗) holds for this series. Finally, there is a version of the theorem without requiring any hypothesis related to semistability coinciding with stability. -Theorem with Positive-Dimensional Stabilisers in U , Giving Theorem 2.12 (U Projective Completions) [6] For a linear H -action on X with respect to an adapted ample linearisation L , there is a sequence of blow-ups along H -invariant (with blow-down map projective subschemes, resulting in a projective scheme X → X) with a linear H -action such that condition (∗ ∗ ∗) is satisfied, and so :X ψ the conclusions of Theorem 2.7 hold. U )s,R/λ(Gm) for the induced linear action of If in addition the stable locus (X// U is nonempty, then this sequence of blowthe reductive group R/λ(Gm ) on X// ups along H -invariant projective subschemes of X can be extended, resulting in  such that condition (∗ ∗ ∗) holds, and such that the another projective scheme X   U )//(R/λ(Gm )) is a geometric quotient of an resulting H -quotient X//H = (X//  open subscheme of X. This theorem provides a non-reductive analogue of the partial desingularisation construction for reductive GIT described at the end of Sect. 2.1. -action on X with respect to an adapted ample lineariDefinition 7 For a linear U sˆ ,U s,U with the sation L , we define Xmin + to be the image of the intersection of X min + under the blow-down map ψ →X :X complement of the exceptional divisor in X given by Theorem 2.12. U is a geometric quotient of the U -action on Since the projective scheme X// s ˆ , U s ˆ , U -action on X the U min + has a geometric quotient Xmin + /U ⊂ X//U , which is invariant under the induced action of R/λ(Gm ). -action already satisfies condition (∗ ∗ ∗), then X = X and thus Note that if the U sˆ ,U s,U the locus Xmin + coincides with the min-stable locus Xmin + given by Definition 6. s,U , X min +

Remark 2.13 As in Theorem 2.9 (cf. Remark 2.10), one first constructs a blow → X by considering stabiliser subgroups for the action of U , and then up X U )s,R/λ(Gm) = ∅ one constructs a further blow-up sequence X  → X by if (X// considering the stabiliser subgroups for the residual action of the reductive subgroup R/λ(Gm ). The slight difference is that in the first step, we look at U (j ) -stabilisers 0 . as well as U -stabilisers for x ∈ Xmin 0 \ U Zmin (X), where Zmin (X) is defined as Zmin for s,U = X Recall that X min +

min

s,U \ E) more precisely, we instead of X. In order to describe Xsˆ ,U = ψ (X X min + min +

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from X in more detail. Let U = U (0) > need to consider the construction of X (1) (s) (s+1) U > ... > U > U = {e} be the derived series for U , as before, and let (j ) = U (j )  λ(Gm ) for 0  j  s + 1; then U (j ) and U (j ) are normal in H . The U first step in the construction is to take the largest j such that (∗ ∗ ∗) does not hold (j ) and then blow X up along the closure of for U (j )

0 : dim StabU (j) (x) = dmax } Cj (X) = {x ∈ Xmin (j )

0 ; we let ψ where dmax is the maximal value of dim StabU (j) (x) for x ∈ Xmin (1) : X(1) → X denote the resulting blow-up, with exceptional divisor denoted E(1) . 0 , and (X )0 is an open Here Cj (X) is an H -invariant closed subscheme of Xmin (1) min 0 ) of X 0 subscheme of the blow-up (ψ(1) )−1 (Xmin min along Cj (X) [6]. Furthermore if x ∈ Zmin and L is very ample then H 0 (X, L) decomposes as a representation of StabU (x) as the direct sum of a trivial one-dimensional representation (which is a weight space for the action of λ(Gm )) and the subspace

{σ ∈ H 0 (X, L) : σx = 0} which is isomorphic as a StabU (x)  λ(Gm )-module to the tangent space to x in (j ) , so that P(H 0 (X, L)∗ ). From this it follows that if (∗ ∗ ∗) does not hold for U 0 0 Cj (X) is a proper closed subscheme of Xmin , then successively blowing up Xmin (j ) 0 along Cj (X) strictly reduces the value of dmax . Thus we can obtain X min from 0 0 Xmin inductively by blowing up Xmin along Cj (X), removing the complement of (X(1) )0min in the result and repeating the process until (∗ ∗ ∗) is satisfied.

sˆ ,U 0 To describe Xmin + , we therefore need to understand Zmin (X(1) ) and (X(1) )min , as well as the centre of the next blow-up. If Zmin ⊆ Cj (X) then Zmin (X(1) ) is the proper transform of Zmin in X(1) and (j )

0 ψ(1) ((X(1) )0min \ E(1)) = {x ∈ Xmin : dim StabU (j) (p(x)) < dmax }.

However if Zmin ⊆ Cj (X) then describing Zmin (X(1) ) and ψ(1) ((X(1) )0min \ E(1) ) is more complicated. In this situation Zmin (X(1) ) ⊆ (ψ(1) )−1 (Zmin ) is the image under p(1) of 0 (ψ(1) )−1 {x ∈ Xmin \Cj (X) : λ(Gm ) acts on the fibre of L∗ over lim λ(t)·x with weight rmin } t→∞

where rmin is minimal among those natural numbers r such that 0 {x ∈ Xmin : λ(Gm ) acts on the fibre of L∗ over lim λ(t) · x with weight r} ⊆ Cj (X). t→∞

Remark 2.14 It follows from Theorem 2.12 and the construction described in sˆ ,U Remark 2.13 that the open H -invariant subscheme Xmin + of X is nonempty unless

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U Zmin is dense in X. Furthermore, by construction, for each 1 ≤ j ≤ s, the function sˆ ,U dim StabU (j) (−) is constant on Xmin +. We will see in the next section that by applying (the proof of) Theorem 2.12 to the closures of subschemes of X where the dimensions of the U (j ) -stabilisers take different values, and combining this with the partial desingularisation construction of [20] for reductive GIT quotients, X can be stratified so that each stratum is a locally closed H -invariant subscheme of X with a geometric quotient by the action of H .

3 Stratifying Quotient Stacks Let H = U  R be a linear algebraic group with internally graded unipotent radical U (cf. Definition 3) acting on a projective scheme X over an algebraically closed field k of characteristic 0 with respect to an ample linearisation L . We fix an invariant inner product on the Lie algebra Lie R of the Levi factor, just as in the construction of the HKKN stratification (cf. Sect. 2.1.1). The aim of this section is to prove Theorem 1.1 stated in Sect. 1. We will prove this result using a recursive argument involving the dimensions of X and of H and the number of irreducible components of X. The idea will be to start by defining a ‘minimum’ stratum, which will be a non-empty H -invariant open subscheme of X, and then proceed recursively. We will assume that H is connected and that X is reduced. Remark 3.1 Let us explain why these assumptions do not involve any significant loss of generality. 1. The HKKN stratification {Sβ | β ∈ B} is usually indexed by a finite subset B of a positive Weyl chamber. Then the strata are not necessarily connected even when X is irreducible, and it is often useful to refine the stratification so that the strata are the connected components of Sβ , or are unions of some but not all of these connected components (cf. [19]). There is a similar ambiguity in the construction of the refined stratification defined in this section: at some points we take connected components, but this is not crucial to the definition. Indeed if we wish to allow the group H to be disconnected then we cannot assume that the strata are connected since they are required to be H -invariant. Then instead of taking connected components (which will be invariant under the component H0 of the identity in H ), we can take their H -sweeps, which will be disjoint unions of at most |H /H0| of these connected components. 2. If X is non-reduced, then we can define the stratification on X by using a positive power of L to define an H -equivariant embedding of X into a projective space Pn , and then take the fibre product of X with the stratification on Pn . Indeed, we will see that the stratification is functorial for equivariant closed immersions (this is essentially the third statement in Theorem 1.1). This follows as the reductive

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notions of GIT (semi)stability are functorial and since in the non-reductive case, we are assuming that H has internally graded unipotent radical. The stable loci when H has internally graded unipotent radical and adapted linearisation are also functorial, as they have Hilbert–Mumford style descriptions (see Definition 6). We note that in the more general non-reductive GIT set up described in Sect. 2.2 the notions of (semi)stability are not functorial, as taking H -invariants is not exact, and so there can be invariants which do not extend to the ambient space. The advantage of assuming that X is reduced is that the complement to the open stratum then has a canonical scheme structure; thus it is easier to recursively define the stratification. Let us describe the recursive construction when H is connected and X is reduced. For each linear action on X of H = U  R with internal grading λ : Gm → R and linearisation L with underlying ample line bundle L, we will first use recursion to define a nonempty H -invariant open subscheme S0 (X, H, λ, L ) of X that admits a geometric quotient S0 (X, H, λ, L )/H ; this will be done by considering seven different cases. After defining the open stratum, we will define the stratification {Sγ |γ ∈ Γ } of X with strata Sγ = Sγ (X, H, λ, L ) and index set Γ = Γ (X, H, λ, L ) by letting X1 , . . . , Xk be the connected components of the projective subscheme of X equal to the complement of S0 (X, H, λ, L ), letting S0,i (X, H, λ, L ) for 1  i  m be the connected components of S0 (X, H, λ, L ), and setting Γ (X, H, λ, L ) := {0} × {1, . . . , m} ∪



{Xj } × Γ (Xj , H, λ, L |Xj ).

(2)

1j k

The strata indexed by (0, i) ∈ Γ (X, H, λ, L ) for 1  i  m are then the connected components of the open subscheme S0 (X, H, λ, L ) constructed using the case by case argument below, whereas the stratum indexed by an element (Xj , γ ) for 1 ≤ j ≤ k and γ ∈ Γ (Xj , H, λ, L |Xj ) is S(Xj ,γ ) (X, H, λ, L ) := Sγ (Xj , H, λ, L |Xj ),

(3)

where the strata Sγ (Xj , H, λ, L |Xj ) are constructed by induction. The partial order on Γ then naturally comes from the partial orders on each Γ (Xj , H, λ, L |Xj ) with (0, i) < (Xj , γ ) for all 1  i  m and 1  j  k and γ ∈ Γ (Xj , H, λ, L |Xj ). Let us now describe how to define S0 (X, H, λ, L ) in the seven different cases. Let U = U (0)  U (1) = [U, U ]  . . .  U (s)  {e} be the derived series of U . Let d

0 = {x ∈ Zmin | dim(StabU (x)) = d0 } Zmin

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0 where d0 is the minimal value of dim(StabU (x)) for x ∈ Zmin . Then Zmin is a d0 nonempty H -invariant open subscheme of Zmin , and U Zmin is a nonempty locally closed subscheme of X.

Case 1 First assume that the central one-parameter subgroup λ : Gm → R of R has at least two distinct weights for the linear action of H on X; equivalently Zmin is a projective subscheme of X with Zmin = X. Under this assumption we have two possibilities to consider. d0 is open in X. Then we define Case 1(a) Suppose that U Zmin

S0 (X, H, λ, L )=U {x ∈ S0 (Zmin , R/λ(Gm ), λ0 , L |Zmin ) | dim(StabUj (x))=dj

for 1  j  s}

where λ0 is the trivial one-parameter subgroup that grades the trivial unipotent radical of the reductive group R/λ(Gm ) and dj := min{dim(StabUj (x)) : x ∈ S0 (Zmin, R/λ(Gm ), λ0 , L |Zmin )}. By induction we can assume that S0 (Zmin , R/λ(Gm ), λ0 , L |Zmin ) is a nonempty R-invariant open subscheme of Zmin with a geometric quotient by the action of R/λ(Gm ) (or equivalently by the action of R, since the central one-parameter subgroup λ(Gm ) of R acts trivially on Zmin ). Indeed, we can construct such an open subscheme as in Case 2 described below. Thus S0 (X, H, λ, L ) is an H -invariant nonempty open subscheme of X, and by the proof of Theorem 2.12 (see [6, Remark 2.10]) it has a geometric quotient S0 (X, H, λ, L )/H ∼ = S0 (Zmin , R/λ(Gm ), λ0 , L |Zmin )/R = S0 (Zmin , R/λ(Gm ), λ0 , L |Zmin )/(R/λ(Gm )).

sˆ ,U 0 Case 1(b) Suppose that U Zmin is not open in X. Then the locus Xmin + defined 0 at Definition 7 is a nonempty H -invariant open subset of Xmin \ U Zmin (by sˆ ,U Remark 2.14), which has a geometric quotient Xmin + /U with a projective completion d



sˆ ,U Xmin + /U ⊆ X//U ,

→ X is the blow-up given by Theorem 2.12, and on which R/λ(Gm ) acts where X linearly with respect to an induced linearisation L (see Remark 2.14). We set

 sˆ ,U sˆ ,U S0 (X, H, λ, L ) = x ∈ Xmin + : U x ∈ S0 (Xmin + /U , R/λ(Gm ), λ0 , L ) . Then S0 (X, H, λ, L ) is an H -invariant nonempty open subscheme of X with a by U , and by the inductive construction a geometric quotient S0 (X, H, λ, L )/U

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geometric quotient )/(R/λ(Gm )) S0 (X, H, λ, L )/H = (S0 (X, H, λ, L )/U by H . Case 2 Now assume that the central one-parameter subgroup λ : Gm → R which grades U acts trivially on X, so the unipotent radical U of H = U  R must also act trivially on X. Then without loss of generality we can assume that H = R is reductive and that λ = λ0 is trivial. Case 2(a) Suppose that the stable locus Xs,R for the linear action of R on X is nonempty. Then we let S0 (X, H, λ, L ) = Xs,R and by classical GIT this has a geometric quotient Xs,R /H = Xs,R /R. Case 2(b) Suppose that the semistable locus Xss,R for the linear action of R on X is empty. Then there is a stratum Sβ from the HKKN stratification for X (associated to our invariant inner product on R) such that β = 0 and Sβ = H Yβss = RYβss ∼ = R ×Pβ Yβss is nonempty and open in X (see Sect. 2.1.1 and Remark 2.5). Then we have the following two subcases to consider. Case 2(b)i) Suppose that Yβss = X. Then by induction on dim X and the number of irreducible components of X, S0 (X, H, λ, L ) = R(S0 (Yβss , Pβ , λβ , L |Y ss )∩Yβss ) ∼ = R×Pβ (S0 (Yβss , Pβ , λβ , L |Y ss )∩Yβss ) β

β

is a nonempty R-invariant (and hence H -invariant) open subscheme of X and has a geometric quotient S0 (X, H, λ, L )/H = S0 (X, H, λ, L )/R ∼ = (S0 (Yβss , Pβ , λβ , L |Y ss )∩Yβss )/Pβ . β

Case 2(b)ii) Suppose that Yβss = X. Then Pβ = R so β defines a rational character of R and corresponds to a nontrivial central one-parameter subgroup λβ : Gm → R. If Zβ = X then the one-parameter subgroup λβ (Gm ) acts nontrivially on X, and thus we can use Case 1 and induction to define S0 (X, H, λ, L ) = S0 (X, R, λβ , L )

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so that it is a nonempty R-invariant (and hence H -invariant) open subscheme of X and has a geometric quotient S0 (X, H, λ, L )/H = S0 (X, R, λβ , L )/R. If Zβ = X then λβ (Gm ) acts trivially on X and we can use induction on the dimension of H to define S0 (X, H, λ, L ) = S0 (X, R/λβ (Gm ), λ0 , L ) which is a nonempty R-invariant (and hence H -invariant) open subscheme of X with geometric quotient S0 (X, H, λ, L )/H = S0 (X, R/λβ (Gm ), λ0 , L )/ (R/λβ (Gm )). Case 2(c) Suppose now that Xs,R = ∅ = Xss,R . Recall from Remark 2.5 that the partial desingularisation construction [20] for the linear action of the reductive group R on X can be applied to Xss,R , although, for simplicity, X was assumed to be irreducible in Remark 2.5, which is no longer the case here. This construction  → X and R-linearisation terminates with a birational projective morphism ψ : X    L for an ample line bundle L on X which is a positive integer multiple of ψ ∗ L ⊗ O(−E) where E is the exceptional divisor and 0 <  n/2} ∪ {0}. If β = 2r − n with r > n/2 then a sequence lies in Zβ = Zβss if and only if it contains r entries equal to [1 : 0] and n − r entries equal to [0 : 1], while Yβ = Yβss consists of sequences with precisely r entries equal to [1 : 0], and finally the HKKN

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stratum Sβ consists   of sequences such that exactly r entries coincide. Thus Zβ , Yβ and Sβ all have nr connected components. The semistable stratum corresponds to β = 0 and consists of sequences in which no point of P1 occurs strictly more than n/2 times. In order to describe the refined stratification, note first that the Uβ -stabilisers in Zβ when β = 2r − n are trivial for n/2 < r < n. Therefore in the refined r,n−r stratification, Sβ decomposes as the disjoint union of S2r−n = SL(2)Zβ , consisting of sequences supported at two points of multiplicity r and n − r respectively, and its r, 7/10 −  [22, 23]. While the GIT approach was successful in constructing the first two steps in the Hassett-Keel program for M g , it quickly became clear that classical Hilbert stability constructions do not produce the next step in the program. On the other hand, heuristic computations predict that already the next step in the Hassett-Keel program, if constructed via GIT, necessitates a stability analysis of 6th Hilbert points of bicanonical curves [30]. A divisor class computation of Eq. (10) similarly suggests the following (perhaps overly optimistic) conjecture: Conjecture 5.3 Let Hilb(Pg−1 )ss p,2 be the locus inside the Hilbert scheme of genus g canonically embedded curves consisting of those curves with a semistable (p, 2)syzygy point. Then Hilb(Pg−1 )ss p,2 // SL(g) ' Proj

 m≥0

%% & && % 4 (g − 1)(g − 2) H0 M g , m 8+ − λ−δ . g g(g − p − 1)

It goes without saying that to prove this conjecture in any particular case, one would need to have a good understanding of GIT stability of syzygy points of canonical curves. Aside from some generic stability results discussed above, our knowledge here is very limited. Whatever understanding we have, it does suffice to work out the first non-trivial variant of Conjecture 5.3 for genus six canonical curves. This result is described in more detail in Sect. 6 below. Remark 5.4 A recent work of Aprodu, Bruno, and Sernesi shows that in genus greater than 10 and gonality greater than 3, the (1, 2)-syzygy point of a canonical curve determines the curve uniquely, unless the curve is bielliptic, in which case the (1, 2)-syzygy point of the curve coincides with that of a cone over an elliptic curve [3, Theorem 1]. Thus, it is natural to expect that for g ≥ 11, the GIT quotient Hilb(Pg−1 )ss 1,2 // SL(g) will be a birational model of Mg in which the hyperelliptic, trigonal, and bielliptic loci are contracted (or flipped).

6 GIT for Syzygies of Canonical Genus Six Curves The first instance where the consideration of syzygy points leads to a genuinely new moduli space is the case of genus 6 curves, which is the smallest genus for which the (1, 2)-syzygy point of a canonical curve is well-defined and non-trivial. What aids the GIT stability analysis here is the beautiful geometry of canonical genus 6 curves, given by a well-known story, which we now recall. A smooth genus 6 curve can be exactly one of the following: hyperelliptic, trigonal, bielliptic, a plane

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quintic, or a quadric section of an anti-canonically embedded degree 5 (possibly singular) del Pezzo in P5 . A generic curve appears only in the last case, and only on a smooth del Pezzo. Quadric sections of singular del Pezzos form a divisor in the moduli space called the Gieseker-Petri divisor D6,4 (we follow the taxonomy of [13] for the Gieseker-Petri divisors; in particular D6,4 is the divisor in M6 of curves with a base-point-free g41 for which the Petri map is not injective). Since a smooth del Pezzo Σ of degree 5 is unique up to an isomorphism, and has a group of automorphisms isomorphic to S5 , there is a distinguished birational model of M6 given by X6 := PH0 (Σ, −2KΣ )/S5 .

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This was the model used by Shepherd-Barron to prove rationality of M6 [35]. It has also reappeared recently in the context of the Hassett-Keel program of M6 , as the ultimate non-trivial log canonical model of M6 [32]. In this section, we reinterpret X6 using GIT of (0, 2) and (1, 2)-syzygy points of canonical genus 6 curves. This allows us to also construct the penultimate log canonical model of M6 , and to realize the contraction of the Gieseker-Petri divisor D6,4 as a VGIT two-ray game. Consider a smooth non-hyperelliptic curve C of genus 6. By Max Noether’s theorem, the canonical embedding of C is a projectively normal degree 10 curve in P5 . We set V := H0 (C, KC ), and identify C with its canonical model in PV ∨ ' P5 . According to Schreyer [34], there are exactly two possible graded Betti tables of C, depending on the Clifford index of C.

6.1 Clifford Index 1 We have Cliff(C) = 1 if and only if C has either g31 (i.e., C is trigonal) or g52 (i.e., C is a plane quintic). In this case the Betti table is: 1 683 386 1 Since dim K1,2 (C) = 3, the (1, 2)-syzygy point of C is not defined, and so we need to analyze only Hilbert points of C. In fact, already the stability of 2nd Hilbert point detects finer aspects of the curve’s geometry.

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Proposition 6.1 Suppose C is a canonically embedded smooth genus six curve with Cliff(C) = 1. Then, with respect to the SL(V )-action, the 2nd Hilbert point   Syz(0,2)(C) ∈ Gr 6, Sym2 V (1) is strictly semistable with dim Stab = 8 if and only if C is a plane quintic. (2) is strictly semistable with dim Stab = 6 if and only if C is trigonal with Maroni invariant 0. (3) is unstable if and only if C is trigonal with positive Maroni invariant. Proof The key observation is that the quadrics containing C cut out a surface S of minimal degree in P5 such that Syz(0,2)(C) = Syz(0,2) (S). The stability analysis of Syz(0,2) (S) is greatly simplified by the fact that S is a rational surface with a large automorphism group. If C is trigonal, then the canonical embedding of C lies on a rational normal surface scroll Sa,4−a in P5 , where a ∈ {1, 2} (see [6, pp.12–13] for a modern exposition of the classical work of Maroni [29] and for a discussion of surface scrolls). The homogeneous ideal of Sa,4−a is generated by the 2 × 2 minors of the following matrix %

x0 x1 · · · xa−1 xa+1 xa+2 · · · x4 x1 x2 · · · xa xa+2 xa+3 · · · x5

&

The Maroni invariant of C is |4 − 2a|, and equals to 0 if and only if the scroll is balanced. In the latter case, S2,2 ' P1 × P1 , embedded by the linear system |OP1 ×P1 (1, 2)|. Since H0 (P1 × P1 , O(1, 2)) is an irreducible representation of SL(2) × SL(2) ⊂ Stab(S2,2 ), we conclude that Syz(0,2) (S2,2 ) = Syz(0,2)(C) is strictly semistable by Theorem 2.4, and has dim Stab = 6. Suppose now C is trigonal with a positive Maroni invariant. Since Syz(0,2)(C) = Syz(0,2)(Sa,4−a ), it suffices to destabilize Syz(0,2)(Sa,4−a ), which is easily done using the oneparameter subgroup acting with weight a − 5 on x0 , . . . , xa , and weight a + 1 on xa+1 , . . . , x5 (cf. [14]). If C is a plane quintic, then it lies on the Veronese surface S = v2 (P2 ) ⊂ P5 . The homogeneous ideal of v2 (P2 ) is generated by the 2 × 2 minors of the following symmetric matrix ⎛

⎞ x0 x1 x2 ⎝ x1 x3 x4 ⎠ . x2 x4 x5 Since the Veronese is embedded into P5 by an irreducible representation of SL(3), we conclude that Syz(0,2)(v2 (P2 )) = Syz(0,2)(C) is strictly semistable by Theorem 2.4, with dim Stab = 8.  )

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6.2 Clifford Index 2 In this case the Betti table of C is: 1 65 56 1 We see that the canonical embedding of C satisfies K1,2 (C) = 0, and so has both a well-defined syzygy point Syz(0,2)(C) ∈ Gr(6, Sym2 V ) and a well-defined syzygy point Syz(1,2) (C) ∈ Gr(5, S(2,1)(V )). Given such a curve C, we can thus associate to it a point h(C) := (Syz(0,2) (C), Syz(1,2)(C)) ∈ H := Gr(6, Sym2 V ) × Gr(5, S(2,1)(V )).

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The SL(V )-linearized semiample cone of H is spanned by the pullbacks pi∗ O(1) of the Plücker line bundles from the two factors. This gives rise to a two-ray VGIT game, whose endpoints correspond to GIT for (0, 2) and (1, 2)-syzygy points, respectively. The (1, 2)-syzygy point of C coincides with the (1, 2)-syzygy point of the unique degree 5 surface S containing C, called the second syzygy scheme of C; see [3]. There are two different possibilities: 1. S is a degree 5 del Pezzo surface, namely, a blow-up of P2 at four possibly infinitely near points, embedded anti-canonically into P5 . 2. S is a cone over an elliptic normal curve of degree 5 in P4 . In both cases h0 (P5 , IS (2)) = 5, and C is a quadric section of S. This shows that the space of quadrics cutting out C is a span of five quadrics cutting out S and another free quadric: H0 (P5 , IC (2)) = H0 (P5 , IS (2)) + C*Q, ⊂ Sym2 V , and the only linear syzygies among the quadrics are those coming from S.

6.2.1 Bielliptic Curves and Elliptic Ribbons In genus 6, bielliptic curves arise as quadric sections of a cone over a genus one smooth curve E embedded into P4 by a complete linear system |L | of a line bundle L ∈ Pic5 (E). A related construction that also produces a genus 6 curve out of the same datum is given by ribbons of Bayer and Eisenbud [5]. Given an elliptic curve

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E and L ∈ Pic5 (E), we can consider an infinitesimal thickening OR of OE by L . More precisely, we consider a short exact sequence of sheaves on E of the form 0 → I → OR → OE → 0, where OR is a sheaf of C-algebras, and I is a sheaf of principal ideals such that I 2 = 0 and I ' L −1 as an OE -module. We denote by R the non-reduced curve with structure sheaf OR supported on E, and call R an elliptic ribbon. By construction, R can be specified as an element of Ext1OE (OE , L −1 ) ' H0 (E, L )∨ . Via the natural Gm -action given by scaling the extension class, every genus 6 elliptic ribbon on E can be isotrivially degenerated to the split ribbon L RE := SpecOE

   OE ⊕ L −1 /( 2 ) .

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The connection between elliptic ribbons and bielliptic curves is the following. If E is embedded into P4 by the complete linear system |L | and Cone(E) is a cone L is isomorphic to the double hyperplane section of Cone(E). over it in P5 , then RE L In particular, RE is an isotrivial specialization of every smooth bielliptic genus 6 quadric section of Cone(E). We are now ready to state the following result: Proposition 6.2 (a) Suppose C is a smooth bielliptic curve. Then Syz(0,2)(C) ∈ Gr(6, Sym2 V ) is strictly semistable, Syz(1,2) (C) ∈ Gr(5, S(2,1)V ) is unstable, and h(C) is unstable with respect to every ample linearization on Gr(6, Sym2 V ) × Gr(5, S(2,1)(V )). (b) Suppose C is a genus 6 elliptic ribbon in P5 . Then Syz(0,2)(C) ∈ Gr(6, Sym2 V ) is strictly semistable. Proof Suppose C is a smooth bielliptic curve. As we have discussed, C is a quadric section of a cone S = Cone(E) over a normal elliptic curve E ⊂ P4 , where E is embedded by a complete linear system |L | of degree 5. Choose a basis x0 , x1 , . . . , x5 of H0 (C, KC )∨ such that the vertex of the cone S is given by x1 = · · · = x5 = 0. Let ρ be the one-parameter subgroup of SL(V ) acting with weight −5 on x0 and weight 1 on each of x1 , . . . , x5 . Then all five quadrics in H0 (P5 , IS (2)) are homogeneous of weight 2, while the smallest weight term of the free quadric Q has weight −10. At the same time, all six syzygies in Syz(1,2) (C) = Syz(1,2)(S) are homogeneous of weight 3. This proves all instability claims. It remains to show that Syz(0,2)(C) is actually semistable. Note that lim ρ(t) · Syz(0,2)(C) = Syz(0,2)(R),

t →0

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L is the canonically embedded genus 6 split ribbon supported on E where R = RE and given by Eq. (14). Thus it suffices to prove that R is semistable. Explicitly, IR = (IS , x02 ) and the one-parameter subgroup ρ introduced above is a subgroup of Stab(R). By Theorem 2.4, it suffices to verify semistability of Syz(0,2) (R) with respect to the centralizer of ρ in SL(6). Let μ be a one-parameter subgroup of this centralizer. Suppose μ acts on x0 by x0 → t w0 x0 . Since Syz(0,2)(R) is strictly semistable with respect to ρ, it will be μ-semistable if and only if it is semistable with respect to the one-parameter subgroup μ := ρ −w0 μ5 . Since μ acts trivially on x0 and acts via a one-parameter subgroup of SL(5) on C*x1 , . . . , x5 ,, we have reduced to verifying μ -semistability of the 2nd Hilbert point of E ⊂ P4 . However, E is an abelian variety, embedded by a complete linear series, and so has a semistable 2nd Hilbert point by [25, Corollary 5.2]. Having established semistability of all canonical split elliptic ribbons of genus 6, we immediately obtain Part (b), as every elliptic ribbon isotrivially degenerates to a split ribbon.

6.2.2 Singular Degree 5 del Pezzo Consider now a degree 5 del Pezzo surface Σ0 with exactly one ordinary double point and no other singularities, which is constructed as follows. Let x, y, z be the standard coordinates on P2 , and let X be the blow-up of P2 at the points p1 := [1 : 0 : 0], p2 := [1 : 1 : 0], p3 := [0 : 1 : 0] and p4 := [0 : 0 : 1]. The first three of these points lie on the line z = 0, and so the strict transform of this line in X is a (−2)-curve. The anti-canonical line bundle of X is globally generated, and defines a morphism to P5 whose image is Σ0 , with the A1 -singularity of Σ0 being precisely the contraction of the (−2)-curve on X. The automorphism group of Σ0 will play an important role in our GIT stability analysis, so we note first that Σ0 admits a Gm -action, induced by the scaling action on P2 that fixes p4 and the line z = 0. We then note that there is also an action of the symmetric group S3 , induced by the action of S3 on P2 permuting the three points p1 , p2 , p3 , and leaving p4 fixed. Proposition 6.3 Let C be a quadric section of Σ0 that does not pass through the singular point of Σ0 . Consider an ample linearization of the SL(V )-action on H given by O(1)  O(β). Then h(C) = (Syz(0,2)(C), Syz(1,2) (C)) is: 1. unstable if β > 4. 2. strictly semistable if β = 4. Proof The natural scaling action of Gm on P2 , given by t · [x : y : z] = [t −1 x : t −1 y : z], extends to X, and gives rise to a one-parameter subgroup ρ of Stab(Σ0 ) ⊂ SL(V ). To understand this subgroup, we begin by choosing a Gm -semi-invariant basis of H0 (X, −KX ): a := xy(x − y), b1 := zx 2, b2 := zxy, b3 := zy 2 , c1 := z2 x, c2 := z2 y.

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Re-normalizing the weights of the action so as to obtain a one-parameter subgroup of SL(6), we see that ρ acts via t · (a, b1 , b2 , b3 , c1 , c2 ) = (t −7 a, t −1 b1 , t −1 b2 , t −1 b3 , t 5 c1 , t 5 c2 ). Recall that S3 acts on Σ0 and note that U1 := *c1 , c2 , is the standard representation of S3 , while U2 := *b1 , b2 , b3 , is its second symmetric power. It follows that H0 (Σ0 , OΣ0 (1)) ' H0 (X, −KX ) is a multiplicity-free representation of Aut(Σ0 ). The quadrics cutting out Σ0 are given by the Pfaffians of the following antisymmetric matrix: ⎛

0 ⎜−z ⎜ 1 ⎜ ⎜−z1 ⎜ ⎝−z2 z0

z1 0 −z4 0 −z2

z1 z4 0 −z5 −z3

z2 0 z5 0 −z3

⎞ −z0 z2 ⎟ ⎟ ⎟ z3 ⎟ ⎟ z3 ⎠ 0

(16)

  A routine computation now shows that ρ acts on det H0 P5 , IΣ0 (2) with weight 2,   and on det H0 P5 , IΣ0 (3) with weight 9. We also record that H0 (Σ0 , OΣ0 (2)) is a homomorphic image of *a 2 ,⊕aU2 ⊕Sym2 U1 ⊕*b12 , b1 b2 , b22 , b2 b3 , b32 ,⊕*b1 c1 , b2 c1 , b2 c2 , b3 c2 ,

(17)

which also shows that ρ acts on H0 (Σ0 , OΣ0 (2)) with weights (−14, −8, −8, −8, 10, 10, 10, −2, −2, −2, −2, −2, 4, 4, 4, 4), and on det H0 (Σ0 , OΣ0 (2)) with weight −2. From this, we conclude that the ρ-weight of Syz(1,2)(Σ0 ) is −3, which shows in particular that Syz(1,2)(Σ0 ) is unstable. Suppose C ⊂ Σ0 is a quadric section of Σ0 , given  by an equation Q(z0 , . . . , z6 ) = 0. Since H0 (C, IC (2)) = H0 (Σ0 , IΣ0 (2)) C*Q,, we see that ρ acts on Syz(0,2)(C) with total weight wρ (Syz(0,2)(C)) = −wρ (Q) + wρ (Syz(0,2)(Σ0 )) ≤ 14 − 2 = 12, with equality holding if and only if the lowest ρ-weight term of Q is z02 if and only if C does not pass through the singularity of Σ0 . Continue with the assumption that C does not pass through the singularity of Σ0 . Set C0 := lim ρ(t) · C, t →0

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where we are taking a flat limit in P5 . Then it is easy to see that IC0 = (z02 , IΣ0 ) = (z02 , z3 z4 −z2 z5 , z2 z4 −z1 z5 , z22 −z1 z3 , z1 z3 −z2 z3 −z0 z5 , z1 z2 −z0 z4 +z1 z3 ). (18) The curve C0 will play a prominent role in what follows. We note some of its properties. To begin, C0 is a canonically embedded Gorenstein curve. Schemetheoretically, it is a union of a fourfold line L0 = V(z4 − z5 , z2 − z3 , z1 − z3 , z0 ) and three double lines L1 = V(z4 , z2 , z1 , z0 ), L2 = V(z3 , z2 , z1 , z0 ), and L3 = V(z5 , z3 , z2 , z0 ). By construction, C0 is also the unique Gm -invariant double hyperplane section of Σ0 that does not pass through the singular point of Σ0 . The statement of the proposition now follows from the following lemma: Lemma 6.4 Consider the linearization O(1)  O(β) on H. Then C0 is polystable for β = 4. Furthermore, C0 is destablized by ρ when β > 4, and is destabilized by ρ −1 when β < 4. Proof By the above computation, we have wρ (Syz(0,2)(C0 )) = wρ (z02 ) + wρ (Syz(0,2) (Σ0 )) = 14 − 2 = 12, wρ (Syz(1,2)(C0 )) = wρ (Syz(1,2)(Σ0 )) = −3. The instability claims follow. It remains to show that C0 is polystable for β = 4. For this we note that C0 is fixed by Aut(Σ0 ), and so in particular Aut(C0 ) = Aut(Σ0 ) ' Gm × S3 . Hence, by Theorem 2.4, it suffices to verify polystability of C0 with respect to those oneparameter subgroup of SL(V ) that commute with Gm × S3 ⊂ Aut(C0 ). Any such one-parameter subgroup acts diagonally on the distinguished basis of V given by Eq. (15) as follows t · (z0 , . . . , z5 ) = (t w0 z0 , t w1 z1 , t w1 z2 , t w1 z3 , t w2 z4 , t w2 z5 ), where w0 + 3w1 + 2w2 = 0. Since the ρ-action stabilizes C0 , it suffices to check the polystability of C0 only with respect to the one-parameter subgroups given by weights (−2, 0, 0, 0, 1, 1) and (2, 0, 0, 0, −1, −1). This is a routine calculation using the fact that the quadrics and syzygies of Σ0 are those satisfied by the Pfaffians of the matrix (16).  )  )

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6.3 The Hassett-Keel Program for M6 In this subsection, we bootstrap the stability results of Sect. 6.2 in order to show that the last two steps in the Hassett-Keel program for M6 are constructed via GIT for syzygy points of canonically embedded curves. Let V = C6 . Recall that H = Gr(6, Sym2 V )×Gr(5, S(2,1)(V )) is equipped with a two-dimensional SL(V )-ample cone, whose elements we denote by O(1)  O(β). For every smooth canonically embedded curve C ⊂ PV ∨ of genus six and Clifford index 2, we have a well-defined point h(C) = (Syz(0,2)(C), Syz(1,2) (C)) ∈ H. Denote by Q ⊂ H the Zariski closure of the locus of all such h(C) in H. We have a natural SL(V )-action on Q, and we denote by Q ss (β) the semistable locus in Q with respect to the linearization O(1)  O(β). We also let G (β) := [Q ss (β)/ SL(V )] be the corresponding GIT quotient stack and denote by G(β) its moduli space. Namely, we have that G(β) = Q ss (β)// SL(V ). We recall that Σ is a smooth degree 5 del Pezzo and Σ0 is a degree 5 del Pezzo with a unique A1 -singularity. Let U ⊂ PH0 (Σ0 , −2KΣ0 ) be the open locus of quadric sections of Σ0 not passing through the singularity. We recall that C0 , given by Eq. (18), is the unique curve with Gm -action in U . We note that, on a smooth del Pezzo, there exists a unique C0 ∈ PH0 (Σ, −2KΣ ) such that C0 isotrivially specializes to C0 . Explicitly, we have C0 = 4F1 + 2F2 + 2F3 + 2F4 , where Fi ’s are (−1)-curves on Σ such that F2 , F3 , F4 meet F1 , and have no other pairwise intersections; C0 is the same curve as described in [32, Proposition 2.6]. We will also need the following counterpart of Proposition 6.3: Lemma 6.5 For every C ∈ PH0 (Σ, −2KΣ ), the syzygy point Syz(1,2)(C) is semistable. Proof The second syzygy scheme of C is Σ, and so Syz(1,2)(C) = Syz(1,2)(Σ). Since Σ is embedded into P5 by an irreducible representation of Aut(Σ) = S5 by [35], the semistability of Syz(1,2) (Σ) follows from Theorem 2.4.  ) Having fixed all the notation, we are ready to state the main result of this section: Theorem 6.6 (Contraction of the Gieseker-Petri Divisor in M6 via VGIT) (1) For β > 4, Q ss (β) = {h(C) | C ∈ PH0 (Σ, −2KΣ )}. Moreover, G (β) ' [PH0 (Σ, −2KΣ )/S5 ] is a Deligne-Mumford stack and G(β) ' X6 ' M6 (α), where α ∈ (16/47, 35/102).

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(2) For β = 4, Q ss (4) = Q ss (4 + ) ∪ {h(C) | C ∈ U }. Moreover, G(4) ' M6 (35/102). (3) For β ∈ (4 − , 4), Q ss (β) = {h(C) | C ∈ U , C = C0 } ∪ {h(C) | C ∈ PH0 (Σ, −2KΣ ), C = C0 }. The stack G (β) is Deligne-Mumford, and we have & % 35 + . G(β) ' M6 102 (4) We have a commutative diagram

(19) 35 Moreover, M6 ( 102 + ) is isomorphic to M6 at the generic point of the Gieseker35 35 Petri divisor D6,4 and M6 ( 102 + ) → M6 ( 102 ) is the contraction of this divisor to 35 [C0 ] ∈ M6 ( 102 ).

Proof A starting point for us is the fact that the generic point of Q lies on a smooth del Pezzo of degree 5. Thus the projection of Q to Gr(5, S(2,1)(V )) consists of the closure of a single orbit SL(V ) · Syz(1,2) (Σ). We have seen in Propositions 6.2 and 6.3 that for β > 4 any point of Q lying over the boundary of this orbit’s closure is unstable, and for β = 4 only the point lying over SL(V ) · Syz(1,2)(Σ0 ) becomes strictly semistable. It follows that β = 4 is the first wall, as β decreases, where the semistability locus changes, and so for β > 4 − , a point in Q is semistable only if its projection to Gr(5, S(2,1)(V )) lies in the union of the two orbits SL(V ) · Syz(1,2) (Σ) ∪ SL(V ) · Syz(1,2)(Σ0 ). (1) For β > 4, we have that Q ss (β) ⊂ {h(C) | C ∈ PH0 (Σ, −2KΣ )}. The nonemptiness of Q ss (β) follows from the semistability of Syz(1,2)(C) given by Lemma 6.5 and the generic semistability of Syz(0,2)(C). Since Σ, and hence its every quadric section, has no infinitesimal automorphisms, every semistable point is stable. Using the properness of the GIT quotient stack G (β), we conclude that in fact we must have an equality Q ss (β) = {h(C) | C ∈ PH0 (Σ, −2KΣ )}.

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It follows that G (β) ' {h(C) | C ∈ PH0 (Σ, −2KΣ )}/ SL(V ) ' PH0 (Σ, −2KΣ )/ Aut(Σ) = X6 .

(2) Since β = 4 is the first wall (as β decreases) where the semistability changes, we have {h(C) | C ∈ PH0 (Σ, −2KΣ )} ⊂ Q ss (4). The inclusion {h(C) | C ∈ U } ⊂ Q ss (4) follows by Proposition 6.3. As we have already discussed, there can be no other semistable points for β = 4. Next we note that C0 is the only point in G (4) \ (G (4 − ) ∪ G (4 + )). The basin of attraction of C0 (that is the locus of points isotrivially degenerating to C0 ) in G (4−) consists of all of U , and the basin of attraction of C0 in G (4+) consists of a single point, C0 ∈ PH0 (Σ, −2KΣ ). It follows (by normality of the GIT quotients) that the open immersion of stacks G (4 + ) ⊂ G (4) induces an isomorphism on the level of their moduli spaces. Namely, G(4 + ) ' G(4). (3) Let C be a generic quadric section of Σ0 that does not pass through the singular point of Σ0 . We are going to prove that h(C) = (Syz(0,2)(C), Syz(1,2)(C)) is stable for 0 < β < 4. To this end, consider a double smooth cubic passing through the points p1 = [1 : 0 : 0], p2 = [1 : 1 : 0], p3 = [0 : 1 : 0] and p4 = [0 : 0 : 1] in P2 . Its strict transform in Σ0 is an elliptic ribbon R of genus 6 that does not pass through the singular point of Σ0 . By Proposition 6.3, h(R) = (Syz(0,2)(R), Syz(1,2)(R)) is semistable with respect to the linearization O(1) O(4). By Proposition 6.2, Syz(0,2)(R) is strictly semistable. It follows that h(R) is semistable with respect to all linearizations O(1)  O(β), where β ∈ (0, 4). To prove that Q ss (β) has no strictly semistable points, we note that for β ∈ (4−, 4) no semistable point admits a Gm -action. Indeed, the smooth del Pezzo Σ has no quadric sections with Gm -action, and the only quadric section of the singular del Pezzo Σ0 is C0 , which is unstable for β < 4. This proves that G (β) is Deligne-Mumford for β ∈ (4 − , 4). It remains to show that every quadric section C of Σ0 that does not pass through the singular point of Σ0 and such that C = C0 is in fact semistable for β ∈ (4 − , 4). This follows from the properness of the GIT quotient stack. Indeed, the open inclusion {h(C) | C ∈ U ∩ Q ss (β)} ⊂ {h(C) | C ∈ U , C = C0 } must induce a birational morphism of projective quotients   {h(C) | C ∈ U ∩ Qss (β)}// SL(V ) → U \ [C0 ] // Aut(Σ0 ) ' P(63 , 125 , 243 , 284 )/S3 ,

where the last identification follows from the explicit description of the Gm action on U ⊂ PH0 (Σ0 , −2KΣ0 ) given by Eq. (17). Since Q ss (β) has no strictly semistable points, this morphism must be an isomorphism and so every element of U \[C0 ] is stable in Q ss (4−). In particular, G (4−) is isomorphic to M 6 at the generic point of the Gieseker-Petri divisor D6,4 . (4) Given our stability results, the existence of the commutative diagram (19) follows from the general theory of VGIT. We finally address the identifications of the GIT quotients G(β) with log canonical models appearing in the HassettKeel program for M6 . Since for all β > 4 − , the stacks G (β) parameterize

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Gorenstein curves with ample dualizing sheaf, and the locus of worse-thannodal curves in G (β) has codimension at least 2, it follows from Eq. (10) (for g = 6 and p = 0, 1) that the Plücker line bundles p1∗ OGr(6,Sym2 V ) (1) and p2∗ OGr(5,S(2,1)(V )) (1) descend to 8λ − δ and 47 2 λ − 3δ, respectively, on the GIT quotient. It follows that O(1)  O(β) descends to G(β) as an ample line bundle % (8λ − δ) + β

& 47 λ − 3δ . 2

(20)

The isomorphism G(4 ∓ ) ' M6 (35/102 ± ) now follows from routine discrepancy computations as in [32, Proposition 4.3], where the identification of X6 with the log canonical models M6 (α) for α ∈ (16/47, 35/102) was first established.  ) Corollary 6.7 The moving slope of M6 is 102/13. Proof By [32, Prop. 4.1], the moving slope of M6 is at most 102/13. It remains to find a family T of curves in M6 passing through the generic point of the GiesekerPetri divisor D6,4 , such that (T · δ0 )/(T · λ) = 102/13, and such that T avoids the boundary divisors δi , i ≥ 1. The existence of such a family is immediate from the fact that the strict transform of D6,4 in M6 (35/102 + ) is the GIT quotient D := (U \ [C0 ])// Aut(Σ0 ) ' P(63 , 125 , 243 , 284 )/S3 . Indeed, since U is an open subset of a complete linear system on Σ0 , it follows that D parameterizes at worst nodal irreducible curves away from codimension 2. Since D has Picard number 1, all curves in D have the same slope. The existence of a requisite family T follows from the fact that the line bundle & 47 λ − 3δ = 102λ − 13δ O(1)  O(4) = (8λ − δ) + 4 2 %

is trivial on D.

 )

6.3.1 Future Directions It is conceivable that one might be able to complete the VGIT analysis of Q ss (β) for all β ∈ (0, 4) ∩ Q. Since genus 6 curves of Clifford index 0 and 1 have no welldefined (1, 2)-syzyzy points, and bielliptic curves are unstable for all positive β by Proposition 6.2, we expect that the resulting GIT quotients G(β) = Q ss (β)// SL(V )

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will parameterize canonical genus 6 curves lying on arbitrary del Pezzo surfaces (in the sense of [10, Definition 8.1.12]) of degree 5 in P5 . By [10, §8.5.1 and Table 8.5], there are just 6 types of singular del Pezzos of degree 5, corresponding to different root sublattices in a root lattice of type A4 , with Σ0 being the least singular of them. We expect that as β decreases, G(β) will contain quadric sections of del Pezzos with worse and worse singularities, until for some small β, we will see all possible del Pezzo surfaces. If for all positive β the GIT quotients G(β) do indeed parameterize only quadric sections of all degree 5 del Pezzos, then, by a standard double covering construction, we will obtain a sequence of compact moduli spaces of K3 surfaces that fits into the Hassett-Keel-Looijenga program (see [26]) for the moduli space of K3 surfaces studied in [4]. Acknowledgements Foremost, this paper owes its existence to the organizers of the Abel Symposium 2017 “Geometry of Moduli,” who gave me an opportunity and motivation to write up this work. I am also indebted to Gavril Farkas, whose influence is evident in every section of this paper, and who generously shared his and Seán Keel’s ideas to use syzygies as the means to construct the canonical model of Mg at an AIM workshop in December 2012. All results in this paper grew out of my attempt to implement these ideas. I am grateful to Anand Deopurkar for his comments and suggestions on an earlier version of this paper. During the preparation of this paper, I was partially supported by the NSA Young Investigator grant H98230-16-1-0061 and Alfred P. Sloan Research Fellowship.

References 1. J. Alper, M. Fedorchuk, D.I. Smyth, Finite Hilbert stability of (bi)canonical curves. Invent. Math. 191(3), 671–718 (2013) 2. M. Aprodu, G. Farkas, Koszul cohomology and applications to moduli, in Grassmannians, Moduli Spaces and Vector Bundles. Clay Math. Proc., vol. 14 (American Mathematical Society, Providence, 2011), pp. 25–50 3. M. Aprodu, A. Bruno, E. Sernesi, A characterization of bielliptic curves via syzygy schemes (2017, Preprint), arXiv:1708.08056 4. M. Artebani, S. Kond¯o, The moduli of curves of genus six and K3 surfaces. Trans. Am. Math. Soc. 363(3), 1445–1462 (2011) 5. D. Bayer, D. Eisenbud, Ribbons and their canonical embeddings. Trans. Am. Math. Soc. 347(3), 719–756 (1995) 6. M. Coppens, The Weierstrass gap sequence of the ordinary ramification points of trigonal coverings of P1 ; existence of a kind of Weierstrass gap sequence. J. Pure Appl. Algebra 43(1), 11–25 (1986) 7. M. Cornalba, J. Harris, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. École Norm. Sup. (4) 21(3), 455–475 (1988) 8. A. Deopurkar, The canonical syzygy conjecture for ribbons. Math. Z. 288(3–4), 1157–1164 (2018) 9. A. Deopurkar, M. Fedorchuk, D. Swinarski, Toward GIT stability of syzygies of canonical curves. Algebr. Geom. 3(1), 1–22 (2016) 10. I.V. Dolgachev, Classical Algebraic Geometry. A Modern View (Cambridge University Press, Cambridge, 2012)

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The Topology of Ag and Its Compactifications Klaus Hulek and Orsola Tommasi

Abstract We survey old and new results about the cohomology of the moduli space Ag of principally polarized abelian varieties of genus g and its compactifications. The main emphasis lies on the computation of the cohomology for small genus and on stabilization results. We review both geometric and representation theoretic approaches to the problem. The appendix provides a detailed discussion of computational methods based on trace formulae and automorphic representations, in particular Arthur’s endoscopic classification of automorphic representations for symplectic groups.

1 Introduction The study of moduli spaces of abelian varieties goes back as far as the late nineteenth century when Klein and Fricke studied families of elliptic curves. This continued in the twentieth century with the work of Hecke. The theory of higher dimensional abelian varieties was greatly influenced by C.L. Siegel who studied automorphic forms in several variables. In the 1980s Borel and others started a systematic study of the topology of locally symmetric spaces and thus also moduli spaces of abelian varieties. From 1977 onwards Freitag, Mumford and Tai proved groundbreaking results on the geometry of Siegel modular varieties. Since then a vast body of literature has appeared on abelian varieties and their moduli. One of the fascinating aspects of abelian varieties is that the subject is at the crossroads of several mathematical fields: geometry, arithmetic, topology and representation theory. In this survey we will restrict ourselves to essentially one

K. Hulek () Institut für Algebraische Geometrie, Leibniz Universität Hannover, Hannover, Germany e-mail: [email protected] O. Tommasi Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Göteborg, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2018 J. A. Christophersen, K. Ranestad (eds.), Geometry of Moduli, Abel Symposia 14, https://doi.org/10.1007/978-3-319-94881-2_6

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aspect, namely the topology of the moduli space Ag of principally polarized abelian varieties and its compactifications. This in itself is a subject which has been covered in numerous research papers and several survey articles. Of the latter we would like to mention articles by van der Geer and Oort [43], Sankaran and the first author [68], van der Geer [41], Grushevsky [54] and van der Geer’s contribution to the Handbook of Moduli [42]. Needless to say that all of these articles concentrate on different aspects and include new results as progress was made. In this article we will, naturally, recall some of the basic ideas of the subject, but we will in particular concentrate on two aspects. One is the actual computation of the cohomology of Ag and its compactifications in small genus. The other aspect is the phenomenon of stabilization of cohomology, which means that in certain ranges the cohomology groups do not depend on the genus. One of our aims is to show how concepts and techniques from such different fields as algebraic geometry, analysis, differential geometry, representation theory and topology come together fruitfully in this field to provide powerful tools and results. In more detail, we will cover the following topics: In Sect. 2 we will set the scene and introduce the moduli space of principally polarized abelian varieties Ag as an analytic space. In Sect. 3 we introduce the tautological ring of Ag . Various compactifications of Ag will be introduced and discussed in Sect. 4, where we will also recall the proportionality principle. In Sect. 5 we shall recall work on L2 cohomology, Zucker’s conjecture and some results from representation theory. This will mostly be a recapitulation of more classical results, but the concepts and the techniques from this section will play a major role in the final two sections of this survey. In Sect. 6 we will treat the computation of the cohomology in low genus in some detail. In particular, we will discuss the cohomology of Ag itself, but also of its various compactifications, and we will treat both singular and intersection cohomology. Finally, stabilization is the main topic of Sect. 7. Here we not only treat the classical results, such as Borel’s stabilization theorem for Ag and its extension by Charney and Lee to the Satake compactification AgSat , but we will also discuss recent work of Looijenga and Chen as well as stabilization of the cohomology for (partial) toroidal compactifications. In the appendix, by Olivier Taïbi, we explain how the Arthur–Selberg trace formula can be harnessed to explicitly compute the Euler characteristic of certain local systems on Ag and their intermediate extensions to AgSat , i.e. L2 -cohomology by Zucker’s conjecture. Using Arthur’s endoscopic classification and an inductive procedure individual L2 -cohomology groups can be deduced. An alternative computation uses Chenevier and Lannes’ classification of automorphic cuspidal representations for general linear groups having conductor one and which are algebraic of small weight. Following Langlands and Arthur, we give details for the computation of L2 -cohomology in terms of Arthur–Langlands parameters, notably involving branching rules for (half-)spin representations. Throughout this survey we will work over the complex numbers C. We will also restrict to moduli of principally polarized abelian varieties, although the same questions can be asked more generally for abelian varieties with other polarizations, as well as for abelian varieties with extra structure such as complex or real

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multiplication or level structures. This restriction is mostly due to lack of space, but also to the fact that, in particular, moduli spaces with non-principal polarizations have received considerably less attention.

2 The Complex Analytic Approach As we said above the construction of the moduli space Ag of principally polarized abelian varieties (ppav) of dimension g can be approached from different angles: it can be constructed algebraically as the underlying coarse moduli space of the moduli stack of principally polarized abelian varieties [34] or analytically as a locally symmetric domain [13]. The algebraic approach results in a smooth Deligne– Mumford stack defined over Spec(Z) of dimension g(g + 1)/2, the analytic construction gives a normal complex analytic space with finite quotient singularities. The latter is, by the work of Satake [90] and Baily–Borel [11] in fact a quasiprojective variety. Here we recall the main facts about the analytic approach. The Siegel upper half plane is defined as the space of symmetric g × g matrices with positive definite imaginary part Hg = {τ ∈ Mat(g × g, C) | τ = t τ, .(τ ) > 0}.

(1)

This is a homogeneous domain. To explain this we consider the standard symplectic form % & 0 1g Jg = . (2) −1g 0 The real symplectic group Sp(2g, R) is the group fixing this form: Sp(2g, R) = {M ∈ GL(2g, R) | t MJ M = J }.

(3)

Similarly we define Sp(2g, Q) and Sp(2g, C). The discrete subgroup Γg = Sp(2g, Z) will be of special importance for us. The group of (complex) symplectic similitudes is defined by GSp(2g, C) = {M ∈ GL(2g, C) | t MJ M = cJ for some c ∈ C∗ }.

(4)

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The real symplectic group Sp(2g, R) acts on the Siegel space Hg from the left by % M=

AB CD

&

: τ → (Aτ + B)(Cτ + D)−1 .

(5)

Here A, B, C, D are g × g matrices. This action is transitive and the stabilizer of the point i1g is 7 % &8 A B Stab(i1g ) = M ∈ Sp(2g, R) | M = . −B A

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The map %

A B −B A

& → A + iB

defines an isomorphism Stab(i1g ) ∼ = U(g) where U(g) is the unitary group. This is the maximal compact subgroup of Sp(2g, R), and in this way we obtain a description of the Siegel space as a homogeneous domain Hg ∼ = Sp(2g, R)/ U(g).

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The involution τ → −τ −1 defines an involution with i1g an isolated fixed point. Hence Hg is a symmetric homogeneous domain. The object which we are primarily interested in is the quotient Ag = Γg \Hg .

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The discrete group Γg = Sp(2g, Z) acts properly discontinuously on Hg and hence Ag is a normal analytic space with finite quotient singularities. This is a coarse moduli space for principally polarized abelian varieties (ppav), see [13, Chapter 8]. Indeed, given a point [τ ] ∈ Ag one obtains a ppav explicitly as A[τ ] = Cg /Lτ , where Lτ is the lattice in Cg spanned by the columns of the (g × 2g)-matrix (τ, 1g ). There are several variations of this construction. One is that one may want to describe (coarse) moduli spaces of abelian varieties with polarizations which are

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non-principal. This is achieved by replacing the standard symplectic form given by Jg by % Jg =

0 D −D 0

& (9)

where D = diag(d1 , . . . , dg ) is a diagonal matrix and the entries di are positive integers with d1 |d2 | · · · |dg . Another variation involves the introduction of level structures. This results in choosing suitable finite index subgroups of Γg . Here we will only consider the principal congruence subgroups of level , which are defined by Γg () = {g ∈ Sp(2g, Z) | g ≡ 1 mod }. The quotient Ag () = Γg ()\Hg

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parameterizes ppav with a level- structure. The latter is the choice of a symplectic basis of the group A[] of -torsion points on an abelian variety A. Recall that A[] ∼ = (Z/Z)2g and that A[] is equipped with a natural symplectic form, the Weil pairing, see [81, Section IV.20]. If  ≥ 3 then the group Γg () acts freely on Hg , see e.g. [93], [13, Corollary 5.1.10], and hence Ag () is a complex manifold (smooth quasi-projective variety). In particular, analogously to the case of Hg discussed in [13, §8.7], for  ≥ 3 the manifold Ag () carries an honest universal family Xg () → Ag (), which can be defined as the quotient Xg () = Γg ()  Z2g \Hg × Cg .

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Here the semidirect product Γg ()  Z2g is defined by the standard action of Sp(2g, Z) on Z2g and the action is given by (M, m, n) : (τ, z) → (M(τ ), ((Cτ + D)t )−1 z + τ m + n)

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for all M ∈ Γg () and m, n ∈ Zg . The map Xg () → Ag () is induced by the projection Hg × Cg → Hg . The universal family Xg () makes sense also for  = 1, 2 if we define it as an orbifold quotient. This allows to define a universal family Xg := Xg (1) → Ag on Ag . A central object in this theory is the Hodge bundle E. Geometrically, this is given by associating to each point [τ ] ∈ Ag the cotangent space of the abelian variety A[τ ] at the origin. This gives an honest vector bundle over the level covers Ag () and an orbifold vector bundle over Ag . In terms of automorphy factors this can be written as E := Sp(2g, Z)\Hg × Cg

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given by M : (τ, v) → (M(τ ), (Cτ + D)v)

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for M ∈ Sp(2g, Z). As we explained above, the Siegel space Hg is a symmetric homogeneous domain and as such has a compact dual, namely the symplectic Grassmannian Yg = {L ⊂ C2g | dim L = g, Jg |L ≡ 0}.

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This is a homogeneous projective space of complex dimension g(g + 1)/2. In terms of algebraic groups it can be identified with Yg = GSp(2g, C)/Q

(16)

where 7% Q=

AB CD

&

8 ∈ GSp(2g, C) | C = 0

(17)

is a Siegel parabolic subgroup. The Siegel space Hg is the open subset of Yg of all maximal isotropic subspaces L ∈ Yg such that the restriction of the form −iJg |L is positive definite. Concretely, one can associate to τ ∈ Hg the subspace spanned by the rows of the matrix (−1g , τ ). The cohomology ring H • (Yg , Z) is very well understood in terms of Schubert cycles. Moreover Yg is a smooth rational variety and the cycle map defines an isomorphism CH• (Yg ) ∼ = H • (Yg , Z) between the Chow ring and the cohomology ring of Yg . For details we refer the reader to van der Geer’s survey paper [42, p. 492]. Since Yg is a Grassmannian we have a tautological sequence of vector bundles 0→E→H →Q→0

(18)

where E is the tautological subbundle, H is the trivial bundle of rank 2g and Q is the tautological quotient bundle. In particular, the fibre EL at a point [L] ∈ Yg is the isotropic subspace L ⊂ C2g . We denote the Chern classes of E by ui := ci (E), which we can think of as elements in Chow or in the cohomology ring. The exact sequence (18) immediately gives the relation (1 + u1 + u2 + . . . + ug )(1 − u1 + u2 − . . . + (−1)g ug ) = 1.

(19)

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Note that this can also be expressed in the form ch2k (E) = 0, k ≥ 1

(20)

where ch2k (E) denotes the degree 2k part of the Chern character. Theorem 1 The classes ui with i = 1, . . . , g generate CH• (Yg ) ∼ = H • (Yg , Z) and all relations are generated by the relation (1 + u1 + u2 + . . . + ug )(1 − u1 + u2 − . . . + (−1)g ug ) = 1. Definition 1 By Rg we denote the abstract graded ring generated by elements ui ; i = 1, . . . , g subject to relation (19). In particular, the dimension of Rg as a vector space is equal to 2g . As a consequence of Theorem 1 and the definition of Rg we obtain Proposition 1 The intersection form on H • (Yg , Z) defines a perfect pairing on Rg . The ring Rg is a Gorenstein ring with socle u1 u2 . . . ug . Moreover there are natural isomorphisms Rg /(ug ) ∼ = Rg−1 . As a vector space Rg is generated by by εi → 1 − εi .

$ i

uεi i with εi ∈ {0, 1} and the duality is given

3 The Tautological Ring of Ag We have already encountered the Hodge bundle E on Ag in Sect. 2. There we 1 defined it as E = π∗ (ΩX ) where π : Xg → Ag is the universal abelian g /Ag variety. As we pointed out this is an orbifold bundle or, alternatively, an honest vector bundle on Ag () for  ≥ 3, as can be seen from the construction of the universal family Xg () → Ag () given in (11). We use the following notation for the Chern classes λi = ci (E).

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We can view these in either Chow or cohomology (with rational coefficients). Indeed, in view of the fact that the group Sp(2g, Z) does not act freely, we will from now on mostly work with Chow or cohomology with rational coefficients. There is also another way in which the Hodge bundle can be defined. Let us recall that it can be realized explicitly as the quotient of the trivial bundle Hg × Cg

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on which the group Sp(2g, Z) acts by % M=

AB CD

& : (τ, v) → (M(τ ), (Cτ + D)v).

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One can also consider this construction in two steps. First, one considers the embedding of Hg into its compact dual Yg = GSp(2g, C)/Q as explained in Sect. 2. Secondly, one can prove that E coincides with the quotient of the restriction to Hg of the tautological subbundle E defined in (18), by the natural Sp(2g, Z)-action. As explained in [41, § 13], this is a special case of a construction that associates with every complex representation of GL(g, C) a homolomorphic vector bundle on Ag . This construction is very important in the theory of modular forms and we will come back to it (in a slightly different guise) in Sect. 4 below. Definition 2 The tautological ring of Ag is the subring defined by the classes λi , i = 1, . . . , g. We will use this both in the Chow ring CH•Q (Ag ) or in cohomology H • (Ag , Q). The main properties of the tautological ring can be summarized by the following Theorem 2 The following holds in CH•Q (Ag ): (i) (1 + λ1 + λ2 + . . . + λg )(1 − λ1 + λ2 − . . . + (−1)g λg ) = 1 (ii) λg = 0 (iii) There are no further relations between the λ-classes on Ag and hence the tautological ring of Ag is isomorphic to Rg−1 . The same is true in H • (Ag , Q). Proof We refer the reader to van der Geer’s paper [40], where the above statements appear as Theorem 1.1, Proposition 1.2 and Theorem 1.5 respectively. This is further discussed in [42].  )

4 Compactifications and the Proportionality Principle The space Ag admits several compactifications which are geometrically relevant. The smallest compactification is the Satake compactification AgSat , which is a special case of the Baily–Borel compactification for locally symmetric domains. Set-theoretically this is simply the disjoint union AgSat = Ag  Ag−1  . . .  A0 .

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It is, however, anything but trivial to equip this with a suitable topology and an analytic structure. This can be circumvented by using modular forms. A modular

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form of weight k is a holomorphic function f : Hg → C % such that for every M ∈ Sp(2g, Z) with M =

AB CD

& the following holds:

f (M(τ )) = det(Cτ + D)k f (τ ). In terms of the Hodge bundle modular forms of weight k are exactly the sections of the k-fold power of the determinant of the Hodge bundle det(E)⊗k . If g = 1, then we must also add a growth condition on f which ensures holomorphicity at infinity, for g ≥ 2 this condition is automatically satisfied. We denote the space of all modular forms of weight k with respect to the full modular group Γg by Mk (Γg ). This is a finite dimensional vector space. Using other representations of Sp(2g, C) one can generalize this concept to vector valued Siegel modular forms. For an introduction to modular forms we refer the reader to [37, 41]. The spaces Mk (Γg ) form a graded ring ⊕k≥0 Mk (Γg ) and one obtains AgSat = Proj ⊕k≥0 Mk (Γg ).

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Indeed, one can take this as the definition of AgSat . The fact that the graded algebra of modular forms is finitely generated implies that AgSat is a projective variety. It contains Ag as a Zariski open subset, thus providing Ag with the structure of a quasi-projective variety. We say that a modular form is a cusp form if its restriction to the boundary of AgSat , by which we mean the complement of Ag in AgSat , vanishes. The space of cusp forms of weight k is denoted by Sk (Γg ). The Satake compactification AgSat is naturally associated to Ag . However, it has the disadvantage that it is badly singular along the boundary. This can be remedied by considering toroidal compactifications Agtor of Ag . These compactifications were introduced by Mumford, following ideas of Hirzebruch on the resolution of surface singularities. We refer the reader to the standard book [10]. Toroidal compactifications depend on choices, more precisely we need an admissible collection of admissible fans. In the case of principally polarized abelian varieties, this reduces to the choice of one admissible fan Σ covering the rational closure Sym2rc (Rg ) of the space Sym2>0 (Rg ) of positive definite symmetric g×g-matrices. To be more precise, an admissible fan Σ is a collection of rational polyhedral cones lying in Sym2rc (Rg ), with the following properties: it is closed under taking intersections and faces, the union of these cones covers Sym2rc (Rg ) and the collection is invariant under the natural action of GL(g, Z) on Sym2rc (Rg ) with the additional property that there are only finitely many GL(g, Z)-orbits of such cones. The construction of such fans is non-trivial and closely related to the reduction theory of quadratic forms.

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There are three classical decompositions (fans) of Sym2rc (Rg ) that have all been well studied and whose associated toroidal compactifications are by now reasonably well understood, namely the second Voronoi, the perfect cone or first Voronoi and the central cone decomposition, leading to the compactifications AgVor , AgPerf and AgCentr respectively. The Voronoi compactification AgVor has a modular interpretation due to Alexeev [2] and Olsson [85]. Indeed, Alexeev introduced the notions of stable semi-abelic varieties and semi-abelic pairs, for which he constructed a moduli stack. It turns out that this is in general not irreducible and that so-called extra-territorial components exist. The space AgVor is the normalization of the principal component of the coarse moduli scheme associated to Alexeev’s functor. In contrast to this, Olsson’s construction uses logarithmic geometry to give the principal component AgVor directly. The perfect cone or first Voronoi compactification AgPerf is very interesting from the point of view of the Minimal Model Program (MMP). Shepherd-Barron [94] has shown that AgPerf is a Q-factorial variety with canonical singularities if g ≥ 5 and that its canonical divisor is nef if g ≥ 12, in other words AgPerf is, in this range, a canonical model in the sense of MMP. We refer the reader also to [5] where some missing arguments from [94] were completed. Finally, the central cone compactification AgCentr coincides with the Igusa blow-up of the Satake compactification AgSat [71]. All toroidal compactifications admit a natural morphism Agtor → AgSat which restrict to the identity on Ag . A priori, a toroidal compactification need not be projective, but there is a projectivity criterion [10, Chapter 4, §2] which guarantees projectivity if the underlying decomposition Σ admits a suitable piecewise linear Sp(2g, Z)-invariant support function. All the toroidal compactifications discussed above are projective. For the second Voronoi compactification AgVor it was only in [2] that the existence of a suitable support function was exhibited. For g ≤ 3 the three toroidal compactifications described above coincide, but in general they are all different and none is a refinement of another. Although toroidal compactifications Agtor behave better with respect to singularities than the Satake compactification AgSat , this does not mean that they are necessarily smooth. To start with, the coarse moduli space of Ag is itself a singular variety due to the existence of abelian varieties with non-trivial automorphisms. These are, however, only finite quotient singularities and we can always avoid these by going to level covers of level  ≥ 3. We refer to this situation as stack smooth. For g ≤ 3 the toroidal compactifications described above are also stack smooth, but this changes considerably for g ≥ 4, when singularities do appear. A priori, the only property we know of these singularities is that they are (finite quotients of) toric singularities. For a discussion of the singularities of AgVor and AgPerf see [29]. By taking subdivisions of the cones we can for each toroidal compactification Agtor obtain a smooth toroidal resolution A˜gtor → Agtor . We shall refer to these compactifications as (stack) smooth toroidal compactifications, often dropping the word stack in this context.

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It is natural to ask whether the classes λi can be extended to AgSat or to toroidal compactifications Agtor . As we will explain later in Sect. 7, it was indeed shown by Charney and Lee [20] that the λ-classes can be lifted to the Satake compactification AgSat via the restriction map H 2i (AgSat , Q) → H 2i (Ag , Q). These lifts are, however, not canonical. Another lift was obtained by Goresky and Pardon [47], working, however, with cohomology with complex coefficients. Their classes 2i Sat are canonically defined and we denote them by λGP i ∈ H (Ag , C). It was recently shown by Looijenga [77] that there are values of i for which the Goresky–Pardon classes have a non-trivial imaginary part and hence differ from the Charney–Lee classes. This will be discussed in more detailed in Theorem 24. The next question is whether the λ-classes can be extended to toroidal compactifications Agtor . By a result of Chai and Faltings [34] the Hodge bundle E can be extended to toroidal compactifications Agtor . The argument is that one can define a universal semi-abelian scheme over Agtor and fibrewise one can then take the cotangent space at the origin. In this way we obtain extensions of the λ-classes in cohomology or in the operational Chow ring. Analytically, Mumford [82] proved ˜ to any smooth toroidal that one can extend the Hodge bundle as a vector bundle E tor tor Sat ˜ ˜ compactification Ag . Moreover, if p : Ag → Ag is the canonical map, then by [47] we have ˜ = p∗ (λGP ). ci (E) i We also note the following: if D is the (reducible) boundary divisor in a level A˜gtor () with  ≥ 3, then by [34, p. 25] ˜ ∼ Sym2 (E) = Ω 1 ˜tor

Ag ()

(D).

˜ on A˜tor also by λi . In order to simplify the notation we denote the classes ci (E) g It is a crucial result that the basic relation (i) of Theorem 2 also extends to smooth toroidal compactifications. Theorem 3 The following relation holds in CH•Q (A˜gtor ): (1 + λ1 + λ2 + . . . + λg )(1 − λ1 + λ2 − . . . + (−1)g λg ) = 1.

(25)

Proof This was shown in cohomology by van der Geer [40] and in the Chow ring by Esnault and Viehweg [31].  ) As before we will define the tautological subring of the Chow ring CH•Q (A˜gtor ) (or of the cohomology ring H • (A˜gtor , Q)) as the subring generated by the (extended) λ-classes. Now λg = 0 and we obtain the following Theorem 4 The tautological ring of A˜gtor is isomorphic to Rg .

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Proof We first note that the relation (25) holds. The statement then follows since the intersection form defines a perfect pairing on the λ-classes. In particular we have ⎞ ⎛ g  1 1 ⎝ (2j − 1)!!⎠ λ 2 g(g+1) = λ1 . . . λg =  0. 1 (g(g + 1))/2)! j =1

 ) Indeed, one can think of the tautological ring as part of the cohomology contained in all (smooth) toroidal compactifications of Ag . Given any two such toroidal compactifications one can always find a common smooth resolution and pull the λ-classes back to this space. In this sense the tautological ring does not depend on a particular chosen compactification A˜gtor . The top intersection numbers of the λ-classes can be computed explicitly by relating them to (known) intersection numbers on the compact dual. This is a special case of the Hirzebruch–Mumford proportionality, which had first been found by Hirzebruch in the co-compact case [66, 67] and then been extended by Mumford [82] to the non-compact case. Theorem 5 The top intersection numbers of the λ-classes on a smooth toroidal compactification A˜gtor are proportional to the corresponding top intersection numbers of the Chern classes of the universal subbundle on the compact dual Yg . # More precisely if ni are non-negative integers with ini = g(g + 1)/2, then ng

λn11 · . . . · λg = (−1)

1 2 g(g+1)

⎛ ⎞ g 1  ng ⎝ ζ (1 − 2j )⎠ un11 · . . . · ug . 2g j =1

As a corollary, see also the proof of Theorem 4, we obtain Corollary 1 1 2 g(g+1)

λ1

= (−1)

1 2 g(g+1)

(g(g + 1)/2)! 2g

!

" g  ζ(1 − 2k) . (2k − 1)!!

k=1

We note that the formula we give here is the intersection number on the stack Ag , i.e. we take the involution given by −1 into account. In particular this means that the degree of the Hodge line bundle on A1Sat equals 1/24. This can also be rewritten in terms of Bernoulli numbers. Recall that the Bernoulli numbers Bj are defined by the generating function ∞

 xk x = Bk , x e −1 k! k=0

|x| < 2π

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and the relation between the Bernoulli numbers and the ζ -function is given by ζ (−n) = (−1)n

Bn+1 n+1

respectively B2n = (−1)n−1

2(2n)! ζ(2n). (2π)2n

A further application of the Hirzebruch–Mumford proportionality is that it describes the growth behaviour of the dimension of the spaces of modular forms of weight k. Theorem 6 The dimension of the space of modular forms of weight k with respect to the group Γg grows asymptotically as follows when k is even and goes to infinity: 1

1

dim Mk (Γg ) ∼ 2 2 (g−1)(g−2) k 2 g(g+1)

g  (j − 1)! (−1)j −1 B2j . (2j )!

j =1

Proof The proof consists of several steps. The first is to go to a level- cover and ˜ ⊗k as the modular forms of weight apply Riemann–Roch to the line bundle (det E) k with respect to the principal congruence subgroup Γg () are just the sections of this line bundle. The second step is to prove that this line bundle has no higher ˜ ⊗k gives the cohomology. Consequently, the Riemann–Roch expression for (det E) dimension of the space of sections, and the leading term (as k grows) is determined 1 g(g+1) on A˜tor (). This shows that by the self-intersection number λ 2 g

1

1

1

1

dim Mk (Γg ()) ∼ 2− 2 g(g+1)−g k 2 g(g+1) [Γg () : Γg ] Vg π − 2 g(g+1) where Vg is Siegel’s volume Vg = 2 g

2 +1

π

1 2 g(g+1)

g  (j − 1)! (−1)j −1 B2j . (2j )!

j =1

The third step is to descend to Ag by applying the Noether–Lefschetz fixed-point formula. It turns out that this does not affect the leading term, with the exception of cancelling the index [Γg () : Γg ].  ) This was used by Tai [97] in his proof that Ag is of general type for g ≥ 9. The same principle can be applied to compute the growth behaviour of the space of modular forms or cusp forms also in the non-principally polarized case, see e.g. [68, Sect. II.2]. Indeed, Hirzebruch–Mumford proportionality can also be used to

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study other homogeneous domains, for example orthogonal modular varieties, see [50, 51].

5 L2 Cohomology and Zucker’s Conjecture In the 1970s and 1980s great efforts were made to understand the cohomology of locally symmetric domains. In the course of this various cohomology theories were studied, notably intersection cohomology and L2 -cohomology. Here we will briefly recall some basic facts which will be of relevance for the discussions in the following sections. One of the drawbacks of singular (co)homology is that Poincaré duality fails for singular spaces. It was one of the main objectives of Goresky and MacPherson to remedy this situation when they introduced intersection cohomology. Given a space X of real dimension m, one of the starting points of intersection theory is the choice of a good stratification X = Xm ⊃ Xm−1 ⊃ · · · ⊃ X1 ⊃ X0 by closed subsets Xi such that each point x ∈ Xi \ Xi−1 has a neighbourhood Nx which is again suitably stratified and whose homeomorphism type does not depend on x. The usual singular k-chains are then replaced by chains which intersect each stratum Xm−i in a set of dimension at most k − i + p(k) where p(k) is the perversity. This leads to the intersection homology groups I Hk (X, Q) and dually to intersection cohomology I H k (X, Q). We will restrict ourselves here mostly to (complex) algebraic varieties where the strata Xi have real dimension 2i, and we will work with the middle perversity, which means that p(k) = k − 1. Intersection cohomology not only satisfies Poincaré duality, but it also has many other good properties, notably we have a Lefschetz theorem and a Kähler package, including a Hodge decomposition. In case of a smooth manifold, or, more generally, a variety with locally quotient singularities, intersection (co)homology and singular (co)homology coincide. The drawback is that intersection cohomology loses some of its functorial properties (unless one restricts to stratified maps) and that it is typically hard to compute it from first principles. Deligne later gave a sheaf-theoretic construction which is particularly suited to algebraic varieties. The main point is the construction of an intersection cohomology complex I CX whose cohomology gives I H • (X, Q). Finally we mention the decomposition theorem which for a projective morphism f : Y → X relates the intersection cohomology of X with that of Y . For an introduction to intersection cohomology we refer the reader to [73]. An excellent exposition of the decomposition theorem can be found in [26]. Although we are here primarily interested in Ag and its compactifications, much of the technology employed here is not special to this case, but applies more generally to hermitian symmetric domains and hence we will now move our discussion into this more general setting. Let G be a connected reductive group

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which, for simplicity, we assume to be semi-simple, and let K be a maximal compact subgroup. We also assume that D = G/K carries a G-equivariant complex structure in which case we speak of a hermitian symmetric space. The prime example we have in mind is G = Sp(2g, R) and K = U (g) in which case G/K = Hg . If Γ ⊂ G(Q) is an arithmetic subgroup we consider the quotient X = Γ \G/K which is called a locally symmetric space. In our example, namely for Γ = Sp(2g, Z), we obtain Ag = Sp(2g, Z)\ Sp(2g, R)/U (g) = Sp(2g, Z)\Hg . As in the Siegel case, we also have several compactifications in this more general setting. The first is the Baily–Borel compactification XBB which for Ag is nothing but the Satake compactification AgSat . As in the Siegel case it can be defined as the proj of a graded ring of automorphic forms, which gives it the structure of a projective variety. Again as in the Siegel case, one can define toroidal compactifications Xtor which are compact normal analytic spaces. Moreover, there are two further topological compactifications, namely the Borel–Serre compactification XBS and the reductive Borel–Serre compactification XRBS [19]. These are topological spaces which do not carry an analytic structure. The space XBS is a manifold with corners that is homotopy equivalent to X, whereas XRBS is typically a very singular stratified space. More details on their construction and properties can be found in [18]. These spaces are related by maps XBS → XRBS → XBB ← Xtor where the maps on the left hand side of XBB are continuous maps and the map on the right hand side is an analytic map. All of these spaces have natural stratifications which are suitable for intersection cohomology. For a survey on this topic we refer the reader to [45] which we follow closely in parts. Another important cohomology theory is L2 -cohomology. For this one considers the space of square-integrable differential forms 7 8   i i Ω(2)(X) = ω ∈ ΩX | ω ∧ ∗ω < ∞, dω ∧ ∗dω < ∞ . This defines the L2 -cohomology groups i (X) = ker d/ im d. H(2)

These cohomology groups are representation theoretic objects and can be expressed in terms of relative group cohomology as follows, see [16, Theorem 3.5]: i H(2) (X) = H i (g, K; L2 (Γ \G)∞ )

(26)

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where L2 (Γ \G)∞ is the module of L2 -functions on Γ \G such that all derivatives by G-invariant differential operators are square integrable. Indeed, this isomorphism holds not only for cohomology with coefficients in C but more generally for cohomology with values in local systems. The famous Zucker conjecture says that the L2 -cohomology of X and the intersection cohomology of XBB are naturally isomorphic. This was proven independently by Looijenga [76] and Saper and Stern [89] in the late 1980s: Theorem 7 (Zucker Conjecture) There is a natural isomorphism i H(2) (X) ∼ = I H i (XBB , C) for all i ≥ 0.

In 2001 Saper [87, 88] established another isomorphism namely Theorem 8 There is a natural isomorphism I H i (XRBS , C) ∼ = I H i (XBB , C) for all i ≥ 0. We conclude this section with two results concerning specifically the case of abelian varieties as they will be relevant for the discussions in the next sections. The following theorem is part of Borel’s work on the stable cohomology of Ag , see [15, Remark 3.8]: Theorem 9 There is a natural isomorphism k (Ag ) ∼ H(2) = H k (Ag , C) for all k < g.

Let us remark that one can also view this theorem as a consequence of Zucker’s conjecture, since H k (Ag , Q) = I H k (Ag , Q) and I H k (XBB , Q) coincide in degree k < g as a consequence of the fact that the codimension of the boundary of X in XBB is g. Finally we notice the following connection between the tautological ring Rg and intersection cohomology of AgSat , see also [55]: Proposition 2 There is a natural inclusion Rg → I H • (AgSat , Q) of graded vector spaces of the tautological ring into the intersection cohomology of the Satake compactification AgSat . Proof By the natural map from cohomology to intersection cohomology we can interpret the (extended) classes λi on AgSat as classes in I H 2i (AgSat ). Via the decomposition theorem we have an embedding I H 2i (AgSat ) ⊂ H 2i (A˜gtor ) where A˜gtor is a (stack) smooth toroidal compactification. Since the classes λi satisfy the relation of Theorem 3 we obtain a map from Rg to I H • (AgSat , Q). Since moreover the intersection pairing defines a perfect pairing on Rg there can be no further relations among the classes λi ∈ I H 2i (AgSat ) and hence we have an embedding Rg → I H • (AgSat , Q).

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Alternatively, one can see this from the isomorphism (26) by looking at i (X) induced by the decomposition of L2 (Γ \G) into the decomposition of H(2) G-representations. Then it is known that the trivial representation occurs with multiplicity one in L2 (Γ \G) and that its contribution coincides with Rg .  ) Remark 1 We mentioned earlier as a motivation for the introduction of the tautological ring that it is contained in the cohomology of all smooth toroidal compactifications A˜gtor of Ag . Proposition 2 provides an explanation for this. Applying the decomposition theorem to the canonical map A˜gtor → AgSat we find that H • (A˜gtor , Q) contains I H • (AgSat , Q) as a subspace, which itself contains the tautological ring Rg .

6 Computations in Small Genus In this section, we consider a basic topological invariant of Ag and its compactifications, namely the cohomology with Q-coefficients. As we work over the field of complex numbers, the cohomology groups will carry mixed Hodge structures (i.e. a Hodge and a weight filtration). We will describe the mixed Hodge structures whenever this is possible because of their geometric significance. In particular, we will denote by Q(k) the pure Hodge structure of Tate of weight −2k. If H is a mixed Hodge structure, we will denote its Tate twists by H (k) := H ⊗ Q(k). We will also denote any extension 0→B →H →A→0 of a pure Hodge structure A by a pure Hodge structure B by H = A + B. Although its geometric and algebraic importance is obvious, the cohomology ring H • (Ag , Q) is completely known in only surprisingly few cases. The cases g = 0, 1 are of course trivial. The cases g = 2, 3 are special in that the locus of jacobians is dense in Ag in these genera. This can be used to obtain information on the cohomology ring H • (Ag , Q) from the known descriptions of H • (Mg , Q) for these values of g. For g = 2 the Torelli map actually extends to an isomorphism from the Deligne–Mumford compactification M 2 to the (in this case canonical) toroidal compactification of A2 . This map identifies A2 with the locus of stable curves of compact type and from this one can easily obtain that the cohomology ring is isomorphic to the tautological ring in this case. In general, however, the cohomology ring of Ag is larger than the tautological ring. This is already the case for g = 3. In [59], Richard Hain computed the rational cohomology ring of A3 using techniques from Goresky and MacPherson’s stratified Morse theory. His result is the following:

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Theorem 10 ([59]) The rational cohomology ring of A3 is isomorphic to Q[λ1 ]/(λ41 ) in degree k = 6. In degree 6 it is given by a 2-dimensional mixed Hodge structure which is an extension of the form Q(−6) + Q(−3). Let us remark that the class of the extension in H 6 (A3 , Q) is unknown. Hain expects it to be given by a (possibly trivial) multiple of ζ(3). For genus up to three, also the cohomology of all compactifications we mentioned in the previous sections is known. For the Satake compactification, this result is due to Hain in [59, Prop. 2 & 3]: Theorem 11 ([59]) The following holds: (i) The rational cohomology ring of A2Sat is isomorphic to Q[λ1 ]/(λ41 ). (ii) The rational cohomology ring of A3Sat is isomorphic to Q[λ1 ]/(λ71 ) in degree k = 6. In degree 6 it is given by a 3-dimensional mixed Hodge structure which is an extension of the form Q(−3)⊕2 + Q. Sat Hain’s approach is based on first computing the cohomology of the link of Ag−1 in AgSat for g = 2, 3 and then using a Mayer–Vietoris sequence to deduce from this the cohomology of AgSat . Alternatively, one could obtain the same result by looking at the natural stratification (23) of AgSat and calculating the cohomology using the Gysin spectral sequence for cohomology with compact support, which in this case degenerates at E2 and yields  Hc• (Ak , Q), g = 2, 3. H • (AgSat , Q) ∼ = 0≤k≤g

Here Hc• (Ag , Q) denotes cohomology with compact support. We recall that Ag is rationally smooth, so that we can obtain Hc• (Ag , Q) from H • (Ag , Q) by Poincaré duality. For g ≤ 3 the situation is easy for toroidal compactifications as well. Let us recall that the commonly considered toroidal compactifications all coincide in this range, so that we can talk about the toroidal compactification in genus 2 and 3. As mentioned above, the compactification A2tor can be interpreted as the moduli space M 2 of stable genus 2 curves, whose rational cohomology was computed by Mumford in [83]. The cohomology of A3tor can be computed using the Gysin long exact sequence in cohomology with compact support associated with its toroidal stratification. Using this, we proved in [69] that the cohomology of A3tor is isomorphic to the Chow ring of A3tor , which is known by [39]. The results are summarized by the following two theorems. Theorem 12 For the toroidal compactification A2tor the cycle map defines an isomorphism CHQ• (A2tor ) ∼ = H • (A2tor , Q). There is no odd dimensional cohomology and the even Betti numbers are given by i 0 2 4 6 bi 1 2 2 1

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Theorem 13 For the toroidal compactification A3tor the cycle map defines an isomorphism CHQ• (A3tor ) ∼ = H • (A3tor , Q). There is no odd dimensional cohomology and the even Betti numbers are given by i 0 2 4 6 8 10 12 bi 1 2 4 6 4 2 1 We also note that due to van der Geer’s results [39] explicit generators of H • (Agtor , Q) for g = 2, 3 are known, as well as the ring structure. The method of computing the cohomology of a (smooth) toroidal compactification using its natural stratification is sufficiently robust that it can be applied to A4Vor as well. The key point here is that, although a general abelian fourfold is not the jacobian of a curve, the Zariski closure of the locus of jacobians in A4Sat is an ample divisor J 4 . In particular, its complement A4Sat \ J4 = A4 \ J4 is an affine variety of dimension 10 and thus its cohomology groups vanish in degree > 10. Hence, there is a range in degrees where the cohomology of compactifications of A4 can be determined from cohomological information on Ag with g ≤ 3 and on the moduli space M4 of curves of genus 4. Using Poincaré duality, this is enough to compute the cohomology of the second Voronoi compactification A4Vor , which is smooth, in all degrees different from the middle cohomology H 10 . However, a single missing Betti number can always be recovered from the Euler characteristic of the space. By the work of Taïbi, see Proposition (3) in the appendix, it is now known that the Euler characteristics e(A4 ) = 9. This allows us to rephrase the results of [70] as follows: Theorem 14 The following holds: (i) The rational cohomology of A4Vor vanishes in odd degree and is algebraic in all even degrees. The Betti numbers are given by i 0 2 4 6 8 10 12 14 16 18 20 bi 1 3 5 11 17 19 17 11 5 3 1 (ii) The Betti numbers of A4Perf in degree ≤ 8 are given by i 0 1 2 3 4 5 6 7 8 b1 1 0 2 0 4 0 8 0 14 and the rational cohomology in this range consists of Tate Hodge classes of weight 2i for each degree i. (iii) The even Betti numbers of A4Sat satisfy the conditions described below: i 0 2 4 6 8 10 12 14 16 18 20 bi 1 1 1 3 3 ≥ 2 ≥ 2 ≥ 2 ≥ 1 1 1

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where all Hodge structures are pure of Tate type with the exception of H 6 (A4Sat , Q) = Q(−3)⊕2 + Q and H 8 (A4Sat , Q) = Q(−4)⊕2 + Q(−1). Furthermore, the odd Betti numbers of A4Sat vanish in degree ≤ 7. Proof (i) is [70, Theorem 1] updated by taking into account the result e(A4 ) = 9. (ii) is obtained from altering the spectral sequence of [70, Table 1] in order to compute the rational cohomology of A4Perf , replacing the first column with the results on the strata of toroidal rank 4 in [70, Theorem 25]. (iii) can be proven by considering the Gysin exact sequence in cohomology with compact support associated with the natural stratification of A4Sat and taking into account that the cohomology of A4 is known in degree ≥ 12 and that it cannot contain classes of Hodge weight 8 by the shape of the spectral sequence of [70, Table 1].  ) Furthermore, in [70, Corollary 3] we prove that H 12 (A4 , Q) is an extension of Q(−9) (generated by the tautological class λ3 λ31 ) by Q(−6). A reasonable expectation for H • (A4 , Q) would be that it coincides with R3 in all other degrees. However, the local system with constant coefficients is not the only one worth while considering for Ag . Indeed, looking at cohomology with values in non-trivial local systems has very important applications, both for arithmetic and for geometric questions. Let us recall that Hg is contractible and thus the orbifold fundamental group of Ag is isomorphic to Sp(2g, Z). Hence representations of Sp(2g, Z) give rise to (orbifold) local systems on Ag . Recall that all irreducible representations of Sp(2g, C) are defined over the integers. As is well known [38, Chapter 17], the irreducible representations of Sp(2g, C) are indexed by their highest weight λ = (λ1 , . . . , λg ) with λ1 ≥ λ2 ≥ . . . ≥ λg . Then for each highest weight λ we can define a local system Vλ , as follows. We consider the associated rational representation ρλ : Sp(2g, Z) → Vλ and define Vλ := Sp(2g, Z)\(Hg × Vλ ) M(τ, v) = (Mτ, ρλ (M)v). Each point of the stack Ag has an involution given by the inversion on the corresponding abelian variety. It is easy to check that this involution acts by (−1)w(λ) on Vλ , where we call w(λ) := λ1 + · · · + λg the weight of the local system Vλ . This implies that the cohomology of all local systems of odd weight is trivial, so that we can concentrate on local systems of even weight. Let us recall that also cohomology with values in local systems carries mixed Hodge structures. Faltings and Chai studied them in [34, Chapter VI] using the BGG-complex and found in this way a very explicit description of all possible steps in the Hodge filtration. The bounds on the Hodge weights are those coming naturally from Deligne’s Hodge theory: Theorem 15 The mixed Hodge structures on the groups H k (Ag , Vλ ) have weights larger than or equal to k + w(λ).

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Remark 2 In practice, the formulas for cohomology of local systems are easier to state in terms of cohomology with compact support. Let us recall that Poincaré duality for Ag implies % & g(g + 1) g(g+1)−• ∗ Hc• (Ag , Vλ ) ∼ (H (A , V ) ⊗ Q − − w(λ) = g λ 2 so that the weights on Hck (Ag , Vλ ) are smaller than or equal to k + w(λ). To understand how cohomology with values in non-trivial local systems behaves, let us consider the case of the moduli space A1 of elliptic curves first. In this case we obtain a sequence of local systems Vk = Symk V1 for all k ≥ 0, and V1 coincides with the local system R 1 π∗ Q for π : X1 → A1 . The cohomology of A1 with coefficients in Vk is known by Eichler–Shimura theory (see [27]). In particular, by work of Deligne and Elkik [30], the following explicit formula describes the cohomology with compact support of A1 : Hc1 (A1 , V2k ) = S[2k + 2] ⊕ Q

(27)

where S[2k + 2] is a pure Hodge structure of Hodge weight 2k + 1 with Hodge decomposition S[2k +2]C = S2k+2 ⊕S2k+2 where S2k+2 and S2k+2 can be identified with the space of cusp forms of weight 2k+2 and its complex conjugate respectively. Formula (27) has been generalized to genus 2 and 3 by work of Bergström, Faber and van der Geer [12, 32, 33]. Using the fact that for these values of g the image of the Torelli map is dense in Ag , they obtain (partially conjectural) formulas for the Euler characteristic of Hc• (Ag , Vλ ) in the Grothendieck group of rational Hodge structures from counts of the number of curves of genus g with prescribed configurations of marked points, defined over finite fields. Let us observe that these formulas in general do not give descriptions of the cohomology groups themselves, for instance because cancellations may occur in the computation of the Euler characteristic. In the case g = 2 a description of the cohomology groups analogue to that in (27) is given by Petersen in [86]. His main theorem is a description of Hc• (A2 , Va,b ) in terms of cusp forms for SL(2, Z) and vector-valued Siegel cusp forms for Sp(4, Z). In particular, for (a, b) = (0, 0) one has that Hck (A2 , Va,b ) vanishes unless we have 2 ≤ k ≤ 4. The simplest case is the one in which we have a > b. In this case, the result for cohomology with rational coefficients is the following: Theorem 16 ([86, Thm. 2.1]) For a > b and a + b even, we have Hc3 (A2 , Va,b ) = Sa−b,b+3 + Sa−b+2 (−b − 1)⊕sa+b+4 + Sa+3 7 Q(−1) if b = 0, ⊕sa+b+4 + Q(−b − 1) + 0 otherwise. 7 Q if b > 0 and a, b even, Hc2 (A2 , Va,b ) = Sb+2 + Q⊕sa−b−2 + 0 otherwise.

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Here • sj,k denotes the dimension of the space of vector-valued Siegel cusp forms for Sp(4, Z) transforming according to Symj ⊗ detk and Sj,k denotes a 4sj,k dimensional pure Hodge structure of weight j + 2k − 3 with Hodge numbers hj +2k−3,0 = hj +k−1,k−2 = hk−2,j +k−1 = h0,j +2k−3 = sj,k ; • sk denotes the dimension of the space of cusp eigenforms for SL(2, Z) of weight k and Sk denotes the corresponding 2sk -dimensional weight k −1 Hodge structure. Furthermore Hck (A2 , Va,b ) vanishes in all other degrees k. The formula for the local systems Va,a with a > 0 is not much more complicated, but it involves also subspaces of cusp eigenforms that satisfy special properties, i.e. not being Saito–Kurokawa lifts, or the vanishing of the central value L(f, 12 ). Moreover, the group Hc2 (A2 , Va,a ) does not vanish in general but has dimension s2a+4 . The proof of the main result of [86] is based on the generally used approach of decomposing the cohomology of Ag into inner cohomology and Eisenstein cohomology. Inner cohomology is defined as H!• (Ag , Vλ ) := Im[Hc• (Ag , Vλ ) → H • (Ag , Vλ )], while Eisenstein cohomology is defined as the kernel of the same map. The Eisenstein cohomology of A2 with arbitrary coefficients was completely determined by Harder [61]. Thus it is enough to concentrate on inner cohomology. • (A , V ⊗ C) is defined as the image in The cuspidal cohomology Hcusp g λ • H (Ag , Vλ ) of the space of harmonic Vλ -valued forms whose coefficients are Vλ valued cusp forms. By [15, Cor. 5.5], cuspidal cohomology is a subspace of inner cohomology. Since the natural map from cohomology with compact support to cohomology factors through L2 -cohomology, we always have a chain of inclusions • • Hcusp (Ag , Vλ ⊗ C) ⊂ H!• (Ag , Vλ ⊗ C) ⊂ H(2) (Ag , Vλ ⊗ C).

These inclusions become more explicit if we consider the Hecke algebra action • (A , V ⊗ C) that comes from its interpretation in terms of (g, K )on H(2) g λ ∞ • (A , V ⊗ C) can be decomposed as direct sum cohomology. In this way H(2) g λ of pieces associated to the elements of the discrete spectrum of L2 (Γ \G). Then • (A , V ⊗ C) corresponds to the pieces of the decomposition corresponding Hcusp g λ to cuspidal forms. Let us recall that one can realize the quotient Ag / ± 1 as a Shimura variety for the group PGSp(2g, Z). The coarse moduli spaces of Ag / ± 1 and Ag coincide and it is easy to identify local systems on both spaces. Moreover, in the case g = 2 there is a very precise description of automorphic forms for PGSp(4, Z) in [35], and in particular of all representations in the discrete spectrum. A careful analysis of these results allows Petersen to prove in [86, Prop. 4.2] that there is an equality • (A , V H!• (A2 , Va,b ) = Hcusp 2 a,b ⊗ C) of the inner and the cuspidal cohomology

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in this case. Moreover, the decomposition of the discrete spectrum of PGSp(4, Z) described by Flicker can be used to obtain an explicit formula for H!• (A2 , Va,b ) as well. We want to conclude this section with a discussion about the intersection cohomology of AgSat and the toroidal compactifications for small genus. For g ≤ 4 a geometric approach was given in [55] where all intersection Betti numbers with the exception of the middle Betti number in genus 4 were determined. As it was pointed out to us by Dan Petersen, representation theoretic methods and in particular the work by O. Taïbi, give an alternative and very powerful method for computing these numbers, as will be discussed in the appendix, in particular Theorem 32. For “small” weights λ, and in particular for λ = 0 and g ≤ 11 the classification theorem due to Chenevier and Lannes [22, Théorème 3.3] is a very effective alternative to compute these intersection Betti numbers, also using Arthur’s multiplicity formula for symplectic groups. Here we state the following Theorem 17 For g ≤ 5 there is an isomorphism of graded vector spaces between the intersection cohomology of the Satake compactification AgSat and the tautological ring I H • (AgSat ) ∼ = Rg . Remark 3 As explained in the appendix, see Remark 6, this result is sharp. For g ≥ 6 there is a proper inclusion Rg  I H • (AgSat ) and starting from g = 9 there is even non-trivial intersection cohomology in odd degree. See also Example 2 for the computation of intersection cohomology for g = 6, 7. Proof There are two possibilities to compute the intersection cohomology of AgSat , at least in principle, explicitly. The first one is geometric in nature, the second uses representation theory. We shall first discuss the geometric approach. This was developed in [55] and is based on the decomposition theorem due to Beilinson, Bernstein, Deligne and Gabber. For this we refer the reader to the excellent survey paper by de Cataldo and Migliorini [26]. We shall discuss this here in the special case of genus 4. We use the stratification of the Satake compactification given by A4Sat = A4  A3  A2  A1  A0 . In this genus the morphism ϕ : A4Vor → A4Sat is a resolution of singularities (up to finite quotients). We denote βi0 := ϕ −1 (A4−i ). Then ϕ|β 0 : βi0 → A4−i is a topological fibration (but the fibres are typically i

not smooth). This is the basic set-up of the decomposition theorem. Since A4Vor is rationally smooth, its cohomology and intersection cohomology coincide. Taking into account that the complex dimension of A4Vor is 10 and that Ak has dimension

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k(k + 1)/2, the decomposition theorem then gives the following (non canonical) isomorphism H m (A4Vor , Q) ∼ = I H m (A4Sat , Q) ⊕



I H m−10+k(k+1)/2+i (AkSat , Li,k,β )

k · · · > λg > 0, then H k (Ag , Vλ1 ,...,λg ) vanishes for k < dimC Ag . The theorem above is a special case of [88, Theorem 5], which holds for all quotients of a hermitian symmetric domain or equal-rank symmetric space. Expectedly, Saper’s techniques can be employed to give better bounds for the vanishing of certain classes of non-regular local systems as well. The fact that cohomology with values in non-regular local systems vanishes implies that in small degree, it is easy to describe the cohomology of spaces that are fibered over Ag . In particular, Theorem 19 implies that the cohomology of the universal family Xg → Ag and that of its fiber products stabilize: Theorem 21 ([56, Thm. 6.1]) For all n, the rational cohomology of the nth fibre product Xg×n of the universal family stabilizes in degree k < g and in this range it is isomorphic to the free H • (A∞ , Q)-algebra generated by the classes Ti := pi∗ (Θ) and Pj k := pj∗k (P ) for i = 1, . . . , n and 1 ≤ j < k ≤ n, where pi : Xg×n → Xg , pj k : Xg×n → Xg×2 are the projections and we denote by Θ ∈ H 2 (Xg , Q) and P ∈ H 2 (Xg×2 , Q) the class of the universal theta divisor and of the universal Poincaré divisor normalized along the zero section, respectively. The next natural question is whether cohomological stability also holds for compactifications of Ag . This is of particular relevance for families of compactifications to which the product maps Pr : Ag1 × Ag2 → Ag1 +g2 extend. The analogue of Question 1 for the Baily–Borel–Satake compactification was settled already in the 1980s by Charney and Lee: Theorem 22 ([20]) The rational cohomology of AgSat stabilizes in degree k < g. in this range, the cohomology is isomorphic to the polynomial algebra Q[x2 , x6 , . . . x4i+2 , . . . ] ⊗ Q[y6 , y10 , . . . y4j +2 , . . . ] generated by classes x4i+2 (i ≥ 0) and y4j +2 (j ≥ 1) of degree 4i + 2 and 4j + 2 respectively. It follows from Charney and Lee’s construction that the classes x4i+2 restrict to λ2i+1 on Ag , whereas the classes y4j +2 vanish on Ag .

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The proof of the theorem above combines Borel’s results about the stable cohomology of Sp(2g, Z) and GL(n, Z) with techniques from homotopy theory. First, it is proved that the rational cohomology of AgSat is canonically isomorphic to the cohomology of the geometric realization of Giffen’s category Wg , which arises from 9 9 hermitian K-theory. The limit for g ≥ ∞ of these geometric realizations 9Wg 9 can then be realized as the base space of a fibration whose total space has rational cohomology isomorphic to H • (Sp(∞, Z), Q) and whose fibres have rational cohomology isomorphic to H • (GL(∞, Z), Q). This immediately yields the description of the generators of the stable cohomology of AgSat . The stability range is proved by looking directly at the stability range for Giffen’s category. In particular, this part of the proof is independent of Borel’s constructions and shows that the stability range for the cohomology of Ag should indeed be k < g. However, because of the fact that Charney and Lee replace AgSat with its Q9 9 homology equivalent space 9Wg 9, the geometric meaning of the x- and y-classes remains unclear. This gives rise to the following two questions: Question 2 What is the geometrical meaning of the classes y4j +2 ? In particular, what is their Hodge weight? Question 3 Is there a canonical way to lift λ4i+2 from H • (Ag , Q) to H • (AgSat , Q) for 4i + 2 < g? The answer to the first question was obtained recently by Chen and Looijenga in [21]. Basically, in their paper they succeed in redoing Charney–Lee’s proof using only algebro-geometric constructions. In particular, they work directly on AgSat rather than passing to Giffen’s category and study its rational cohomology by investigating the Leray spectral sequence associated with the inclusion Ag → AgSat . The E2 -term of this spectral sequence can be described explicitly using the fact that each point in a stratum Ak ⊂ AgSat has an arbitrarily small neighbourhood which is a virtual classifying space for an arithmetic subgroup Pg (k) ⊂ Sp(2g, Z) which is fibered over GL(g − k, Z). In the stable range this can be used to construct a spectral sequence converging to H • (Ag , Q) with E2 -terms isomorphic to those in the spectral sequence considered by Charney and Lee. Furthermore, this algebrogeometric approach allows to describe explicitly the Hodge type of the y-classes. This is done by first giving a “local” interpretation of them as classes lying over the cusp A0 of AgSat , and then using the existence of a toroidal compactification to describe the Hodge type of the y-classes in the spirit of Deligne’s Hodge theory [28]. This gives the following result: Theorem 23 ([21, Theorem 1.2]) The y-classes have Hodge type (0, 0). Concerning Question 3, it is clear that while the y-classes are canonically defined, the x-classes are not. On the other hand, Goresky and Pardon [47] defined ∈ H • (AgSat , C). As canonical lifts of the λ-classes from λi ∈ H • (Ag , C) to λGP i the pull-back of x4i+2 and of λGP 2i+1 to any smooth toroidal compactification of Ag coincide, one wonders whether the two classes may coincide. This question was settled in the negative by Looijenga in [77], who studied the properties of

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the Goresky–Pardon classes putting them in the context of the theory of stratified spaces. We summarize the main results about stable cohomology from [77] as follows Theorem 24 ([77]) For 2 < 4r + 2 < g, the Goresky–Pardon lift of the degree 2r + 1 Chern character of the Hodge bundle has a non-trivial imaginary part and its real class lies in H 4r+2 (AgSat , Q). In particular, the classes λGP 2r+1 are different from the x4r+2 in Theorem 23. This is related to an explicit description of the Hodge structures on a certain subspace of stable cohomology. Let us recall that an element x of a Hopf algebra is called primitive if its coproduct satisfies Δ(x) = x ⊗ 1 + 1 ⊗ x. If one considers the Hopf algebra structure of stable cohomology of AgSat , the primitive part in degree 4r + 2 is generated by y4r+2 and the Goresky–Pardon lift chGP 2r+1 of the Chern GP character, which is a degree 2r + 1 polynomial in the λj with j ≤ 2r + 1. The proof of Theorem 24 is based on an explicit computation of the Hodge structures on this primitive part [77, Theorem 5.1] obtained by using the theory of the Beilinson regulator and the explicit description of the y-generators given in [21]. This amounts to describing H 4r+2(AgSat , Q)prim as an extension of a weight 4r+2 Hodge structure generated by chGP 2r+1 by a weight 0 Hodge structure, generated by y4r+2 . Chen and Looijenga’s explicit construction of y2r+2 also yields a construction of a homology class z ∈ H4r+2 (AgSat , Q) that pairs non-trivially with y4r+2 . Hence, one can describe the class of the extension by computing the pairing of chGP 2r+1 with z. This computation is only up to multiplication by a non-zero rational number because of an ambiguity in the definition of z, but it is enough to show that the class of the extension is real and a non-zero rational multiple of ζ(2r+1) . π 2r+1 Let us mention that the question about stabilization is settled also for the reductive Borel–Serre compactification Ag RBS of Ag . Let us recall from Sect. 5 that its cohomology is naturally isomorphic to the L2 -cohomology of Ag and to the intersection homology of AgSat . Combining this with Borel’s Theorems 19 and 9 we can obtain Theorem 25 The intersection cohomology I H k (AgSat , Q) stabilizes in degree k < g to the graded vector space Q[λ1 , λ3 , . . . ]. At this point we would like to point out that the range in which intersection cohomology is tautological given in this theorem can be improved considerably, namely to a wider range k < 2g − 2, and also extended to non-trivial local systems Vλ , for which the intersection cohomology vanishes in the stable range. For details we refer to Theorem 33 in the appendix. The analogue of Question 1 for toroidal compactifications turns out to be a subtle question, which in this form remains open. We dealt with stability questions in a series of papers [56, 57], joint with Sam Grushevsky. Let us recall that toroidal compactifications come in different flavours. The first question to answer is which choice of toroidal compactification is suitable in order to obtain stabilization phenomena in cohomology. At a theoretical level, this

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requires to work with a sequence of compactifications {Ag Σg } where each Σg is an admissible fan in Sym2rc (Rg ). Then the system of maps Ag → Ag+1 extends to the compactification if and only if {Σg } is what is known as an admissible collection of fans (see e.g. [57, Def. 8]). If one wants to ensure that the product maps Ag1 × Ag2 → Ag1 +g2 extend, one needs a stronger condition, which we shall call additivity, namely that the direct sum of a cone σ1 ∈ Σg1 and a cone σ2 ∈ Σg2 should always be a cone in Σg1 +g2 . All toroidal compactifications we mentioned so far are additive. However, only the perfect cone compactification is clearly a good candidate for stability. For instance, the second Voronoi decomposition is ruled out because the number of boundary components of AgVor increases with g, so that the same should happen with H 2 (AgVor , Q). Instead, the perfect cone compactification has the property that for all k ≤ g, the preimage of Ag−k ⊂ AgSat in AgPerf always has codimension k. In particular, the boundary of AgPerf is always irreducible. Let us recall that a toroidal compactification associated to a fan Sym2rc (Rg ) is the disjoint union of locally closed strata βg (σ ) corresponding to the cones σ ∈ Σ up to GL(g, Z)-equivalence. Each stratum βg (σ ) has codimension equal to dimR σ , by which we mean the dimension of the linear space spanned by σ . The rank of a cone σ is defined as the minimal k such that σ is a cone in Σk . If the rank is k, then βg (σ ) maps surjectively to Ag−k under the forgetful morphism to AgSat . The properties of the perfect cone decomposition can be rephrased in terms of the fan by saying that if a cone σ in the perfect cone decomposition has rank k, then its dimension is at least k. Moreover, the number of distinct GL(g, Z)orbits of cones of a fixed dimension  ≤ g is independent of g. This means that the combinatorics of the strata βg (σ ) of codimension  ≤ k is independent of g provided g ≥ k holds. Furthermore, studying the Leray spectral sequence associated to the fibration βg (σ ) → Ag−r with r = rank σ allows to prove that the cohomology of βg (σ ) stabilizes for k < g − r − 1; this stable cohomology consists of algebraic classes and can be described explicitly in terms of the geometry of the cone σ (see [56, Theorem 8.1]). The basic idea of the proof is analogous to the one used in Theorem 21 to describe the cohomology of Xgn . All this suggests that AgPerf is a good candidate for stability. In practice, however, the situation is complicated by the singularities of AgPerf . Let us review the main results of [56, 57] in the case of an arbitrary sequence {AgΣ } of partial toroidal compactifications of Ag associated with an admissible collection of (partial) fans Σ = {Σg }. Here we use the word partial to stress the fact that we don’t require the union of all σ ∈ Σg to be equal to Sym2rc (Rg ). In other words, we are also considering the case in which AgΣ is the union of toroidal open subsets of a (larger) toroidal compactification. Theorem 26 Assume that Σ is an additive collection of fans and that each cone σ ∈ Σg of rank at least 2 satisfies dimR σ ≥

rank σ + 1. 2

(30)

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Then if Σ is simplicial we have that H k (AgΣ , Q) stabilizes for k < g and that cohomology is algebraic in this range, in the sense that cohomology coincides with the image of the Chow ring. Furthermore, there is an isomorphism Σ , Q) ∼ H 2•(A∞ = H 2•(A∞ , Q) ⊗Q Sym• (VΣ )

of free graded algebras between stable cohomology and the algebra over H 2• (A∞ , Q) = Q[λ1 , λ3 , . . . ] generated by the symmetric algebra of the graded vector space spanned by the tensor products of forms in the Q-span of each cone σ that are invariant under the action of the stabilizer Aut σ of σ , for all cones σ that are irreducible with respect to the operation of taking direct sums, i.e. VΣ2k :=





Symk−dimR σ (Q-span of σ )

Aut σ

[σ ]∈[Σ] [σ ] irreducible w.r.t. ⊕

where [Σ] denotes the collection of the orbits of cones in Σ under the action of the general linear group. As we already remarked, the perfect cone compactification satisfies a condition which is stronger than (30), so any rationally smooth open subset of AgPerf which is defined by an additive fan satisfies the assumptions of the theorem. For instance, this implies that the theorem above applies to the case where AgΣ is the smooth locus of AgPerf or the locus where AgPerf is rationally smooth. A more interesting case that satisfies the assumptions of the theorem is the matroidal partial compactification AgMatr defined by the fan Σ Matr of cones defined starting from simple regular matroids. This partial compactification was investigated by Melo and Viviani [79], who showed that the matroidal fan coincides with the intersection ΣgMatr = ΣgPerf ∩ ΣgVor of the perfect cone and second Voronoi fans, so that AgMatr is an open subset in both AgPerf and AgVor . Its geometrical significance is also related to the fact that the image of the extension of the Torelli map to the Deligne–Mumford stable curves is contained in AgMatr , as shown by Alexeev and Brunyate in [3]. Furthermore, by [57, Prop. 19], rational cohomology stabilizes also without the assumption that Σ be additive. In this case stable cohomology is not necessarily a free polynomial algebra, but it still possesses an explicit combinatorial description as a graded vector space. If Σ is not necessarily simplicial, it is not known whether cohomology stabilizes in small degree. However, one can prove that cohomology (and homology) stabilize in small codegree, i.e. close to the top degree g(g + 1). The most natural way to phrase this is to look at Borel–Moore homology of AgΣ , because this is where the cycle map from the Chow ring of AgΣ takes values if AgΣ is singular and possibly non-compact. (If AgΣ is compact, then Borel–Moore homology coincides with the usual homology).

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If Σ is additive, it is possible to prove (see [57, Prop. 9]) that the product maps extend, after going to a suitable level structure, to a transverse embedding AgΣ1 × AgΣ2 → AgΣ1 +g2 . In particular, taking products with a chosen point E ∈ A1 defines Σ . It makes sense to a transverse embedding (in the stacky sense) of AgΣ in Ag+1 wonder in which range the Gysin maps Σ H¯ (g+1)(g+2)−k (Ag+1 , Q) → H¯ g(g+1)−k (AgΣ , Q)

are isomorphisms, where H¯ denotes Borel-Moore homology. Then Theorem 26 generalizes to Theorem 27 If Σ is an additive collection of (partial) fans such that each cone of rank ≥ 2 satisfies (30), then the Borel–Moore homology of AgΣ stabilizes in codegree k < g and the stable homology classes lie in the image of the cycle map. Furthermore, there is an isomorphism of graded vector spaces between stable Borel–Moore homology in codegree 2k and the degree 2k part of H 2• (A∞ , Q) ⊗Q Sym• (VΣ ), where VΣ is defined as in Theorem 26. Let us remark that Borel–Moore homology in small codegree does not have a ring structure a priori. As the assumptions of the theorem above are satisfied by the perfect cone compactification, we get the following result. Corollary 2 ([56, Theorems 1.1 & 1.2]) The rational homology and cohomology of AgPerf stabilize in small codegree, i.e. in degree g(g + 1) − k with k < g. In this range, homology is generated by algebraic classes. As explained in Dutour-Sikiri´c appendix to [57], the state of the art of the classification of orbits of matroidal cones and perfect cone cones is enough to be able to compute the stable Betti numbers of AgMatr in degree up to 30 and the stable Betti numbers of AgPerf in codegree at most 22 (where the result for codegree 22 is actually a lower bound, see [57, Theorems 4 & 5]). Concluding, we state two problems that remain open at the moment. Question 4 Does the cohomology of AgPerf stabilize in small degree k < g? As we have already observed, the answer to Question 4 is related to the behaviour of the singularities of AgPerf and a better understanding of them may be necessary to answer this question. A different question arises as follows. Since stable homology of AgPerf is algebraic, stable homology classes of degree g(g + 1) − k can be lifted (noncanonically) to intersection homology of the same degree. A natural question to ask is whether all intersection cohomology classes in degree k < g are of this form, which we may rephrase using Poincaré duality as

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Question 5 Is there an isomorphism Hg(g+1)−k (AgPerf , Q) ∼ = I H k (AgPerf , Q) for all k < g? Acknowledgements We are grateful to Dan Petersen for very useful comments on an earlier draft of this paper. The second author would like to acknowledge support from her Research Award 2016 of the Faculty of Science of the University of Gothenburg during the preparation of this paper. The first author is grateful to the organizers of the Abel symposium 2017 for a wonderful conference.

Appendix: Computation of Intersection Cohomology Using the Langlands Program Olivier Taïbi CNRS, Unité de mathématiques pures et appliquées, ENS de Lyon, France; [email protected] In this appendix we explain a method for the explicit computation of the Euler characteristics for both the cohomology of local systems Vλ on Ag and their intermediate extensions to AgSat . Furthermore, we explain how to compute individual intersection cohomology groups in the latter case. The main tools here are trace formulas and results on automorphic representations, notably Arthur’s endoscopic classification of automorphic representations of symplectic groups [9]. We start by explaining in Proposition 3 the direct computation of e(A4 ) = 9. This Euler characteristic, as well as e(Ag , Vλ ) for g ≤ 7 and any λ, can be obtained as a byproduct of computations explained in the first part of [98], which focused on L2 -cohomology. In fact by Proposition 4 these are given by the conceptually simple formula (32). The difficulty in evaluating this formula resides in computing certain coefficients, called masses, for which we gave an algorithm in [98]. The number e(A4 ) was missing in [70] to complete the proof of Theorem 14. Next we recall from [98] that the automorphic representations for Sp2g contributing to I H • (AgSat , Vλ ) can be reconstructed from certain sets of automorphic representations of general linear groups, which we shall introduce in Definition 4. Thanks to Arthur’s endoscopic classification [9] specialized to level one and the identification by Arancibia et al. [4] of certain Arthur-Langlands packets with the concrete packets previously constructed by Adams and Johnson [1] in the case of the symplectic groups, combined with analogous computations for certain special orthogonal groups, we have computed the cardinalities of these “building blocks”. Again, this is explicit for g ≤ 7 and arbitrary λ. For g ≤ 11 and “small” λ, the classification by Chenevier and Lannes in [22] of level one algebraic automorphic representations of general linear groups over Q having “motivic weight” ≤ 22 (see Theorem 31 below) gives another method to compute these sets. Using either method, we deduce I H 6 (A3Sat , V1,1,0 ) = 0 in Corollary 3, which was a missing

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ingredient to complete the computation in [55] of I H • (A4Sat , Q) (case g = 4 in Theorem 17). In fact using the computation by Vogan and Zuckerman [102] of the (g, K)-cohomology of Adams-Johnson representations, including the trivial representation of Sp2g (R), we can prove that the intersection cohomology of AgSat is isomorphic to the tautological ring Rg for all g ≤ 5 (see Theorem 17), again by either method. One could deduce from [102] an algorithm to compute intersection cohomology also in the cases where there are non-trivial representations of Sp2g (R) contributing to I H • (AgSat , Vλ ), e.g. for all g ≥ 6 and Vλ = Q. Instead of pursuing this, in Sect. 7 we make explicit the beautiful description by Langlands and Arthur of L2 cohomology in terms of the Archimedean Arthur-Langlands parameters involved, i.e. Adams-Johnson parameters. In fact the correct way to state this description would be to use the endoscopic classification of automorphic representations for GSp2g . Although this classification is not yet known, we can give an unconditional recipe in the case of level one automorphic representations. We conclude the appendix with the example of the computation of I H • (AgSat , Q) for g = 6, 7, and relatively simple formulas to compute the polynomials 

T k dim I H k (AgSat , Vλ )

k

for all values of (g, λ) such that the corresponding set of substitutes for ArthurLanglands parameters of conductor on is known (currently g ≤ 7 and arbitrary λ and all pairs (g, λ) with g + λ1 ≤ 11). Let us recall that for n = 3, 4, 5 mod 8 there is a (unique by [52, Proposition 2.1]) reductive group G over Z such that GR ' SO(n−2, 2). Such G is a special orthogonal group of a lattice, for example E8 ⊕H ⊕2 where H is a hyperbolic lattice. If K is a maximal compact subgroup of G(R), we can also consider the hermitian locally symmetric space G(Z)\G(R)/K 0 = G(Z)0 \G(R)0 /K 0 where G(R)0 (resp. K 0 ) is the identity component of G(R) (resp. K) and G(Z)0 = G(Z) ∩ G(R)0 . Then everything explained in this appendix also applies to this situation, except for the simplification in Proposition 4 which can only be applied to the simply connected cover of G. Using [23, §4.3] one can see that this amounts to considering (son , SO(2) × SO(n − 2))-cohomology instead of (son , S(O(2) × O(n − 2)))-cohomology as in [98], and this simply multiplies Euler characteristics by 2. In fact the analogue of Sect. 7 is much simpler for special orthogonal groups, since they do occur in Shimura data (of abelian type). To be complete we mention that Arthur’s endoscopic classification in [9] is conditional on several announced results which, to the best of our knowledge, are not yet available (see [98, §1.3]).

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Evaluation of a Trace Formula Our first goal is to prove the following result. Proposition 3 We have e(A4) = 9. This number is a byproduct of the explicit computation in [98] of e(2)(Ag , Vλ ) :=

 i (−1)i dimR H(2) (Ag , Vλ )

(31)

i

for arbitrary irreducible algebraic representations Vλ of Sp2g . Each representation Vλ is defined over Q, and L2 -cohomology is defined with respect to an admissible • (A , V ) is a graded real vector space inner product on R ⊗Q Vλ . In particular H(2) g λ (for arbitrary arithmetic symmetric spaces the representation Vλ may not be defined over Q and so in general L2 -cohomology is only naturally defined over C). Recall that by Zucker’s conjecture (31) is also equal to  (−1)i dim I H i (AgSat , Vλ ). i

To evaluate the Euler characteristic (31) we use Arthur’s L2 -Lefschetz trace formula [6]. This is a special case since this is the alternating trace of the unit in the unramified Hecke algebra on these cohomology groups. Arthur obtained this formula by specializing his more general invariant trace formula. In general Arthur’s invariant trace formula yields transcendental values, but for particular functions at the real place (stable sums of pseudo-coefficients of discrete series representations) Arthur obtained a simplified expression all of whose terms can be seen to be rational. Goresky et al. [46, 49] gave a different proof of this formula, by a topological method. In fact they obtained more generally a trace formula for weighted cohomology [48], the case of a lower middle or upper middle weight profile on a hermitian locally symmetric space recovering intersection cohomology of the Baily-Borel compactification. We shall also use split reductive groups over Q other than Sp2g below, which will be of equal rank at the real place but do not give rise to hermitian symmetric spaces. For this reason it is reassuring that Nair [84] proved that in general weighted cohomology groups coincide with Franke’s weighted L2 cohomology groups [36] defined in terms of automorphic forms. The case of usual L2 cohomology corresponds to the lower and upper middle weight profiles in [48]. In particular Nair’s result implies that [49] is a generalization of [6]. In our situation the trace formula can be written  e(2)(Ag , Vλ ) = T (Sp2g , M, λ) M

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where the sum is over conjugacy classes of Levi subgroups M of Sp2g which are R-cuspidal, i.e. isomorphic to GLa1 × GLc2 × Sp2d with a + 2c + d = g. The right hand side is traditionally called the geometric side, although this terminology is confusing in the present context. The most interesting term in the sum is the elliptic part Tell (Sp2g , λ) := T (Sp2g , Sp2g , λ) which is defined as Tell (Sp2g , λ) =



mc tr (c | Vλ ) .

(32)

c∈C(Sp2g )

Here C(Sp2g ) is the finite set of torsion R-elliptic elements in Sp2g (Q) up to conjugation in Sp2g (Q). These can be simply described by certain products of degree 2n of cyclotomic polynomials. The rational numbers mc are “masses” (in the sense of the mass formula, so it would be more correct to call them “weights”) computed adelically, essentially as products of local orbital integrals (at all prime numbers) and global terms (involving Tamagawa numbers and values of certain Artin L-functions at negative integers). We refer the reader to [98] for details. Let us simply mention that for c = ±1 ∈ C(Sp2g ), the local orbital integrals are all equal to 1, and mc is the familiar product ζ (−1)ζ(−3) . . . ζ(1 − 2g). The appearance of other terms in Tell (Sp2g , λ) corresponding to non-central elements in C(Sp2g ) is explained by the fact that the action of Sp2g (Z)/{±1} on Hg is not free. To evaluate Tell (Sp2g , λ) explicitly, the main difficulty consists in computing the local orbital integrals. An algorithm was given in [98, §3.2]. In practice these are computable (by a computer) at least for g ≤ 7. For g = 2 they were essentially computed by Tsushima in [101]. For g = 3 they could also be computed by a (dedicated) human being. See [98, Table 9] for g = 3, and [100] for higher g. The following table contains the number of masses in each rank, taking into account that m−c = mc . g  card C(Sp2g )/{±1} 

1 3

2 3 4 5 6 7 12 32 92 219 530 1158

In general the elliptic part of the geometric side of the L2 -Lefschetz trace formula does not seem to have any spectral or cohomological meaning, but for unit Hecke operators and simply connected groups, such as Sp2g , it turns out that it does. Proposition 4 Let G be a simply connected reductive group over Q. Assume that GR has equal rank, i.e. GR admits a maximal torus (defined over R) which is anisotropic, and that GR is not anisotropic, i.e. G(R) is not compact. Let Kf be a compact open subgroup of G(Af ) and let Γ = G(Q) ∩ Kf . Let K∞ be a maximal compact subgroup of G(R) (which is connected). Then for any irreducible algebraic representation Vλ of G(C), in Arthur’s L2 -Lefschetz trace formula e(2) (Γ \G(R)/K∞ , Vλ ) =

 M

T (G, Kf , M, λ)

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where the sum is over G(Q)-conjugacy classes of cuspidal Levi subgroups of G, the elliptic term Tell (G, Kf , λ) := T (G, Kf , G, λ) is equal to ec (Γ \G(R)/K∞ , Vλ ) =

 (−1)i dim Hci (Γ \G(R)/K∞ , Vλ ). i

Proof Note that by strong approximation (using that G(R) is not compact), the natural inclusion Γ \G(R)/K∞ → G(Q)\G(Af )/K∞ Kf is an isomorphism between orbifolds. We claim that in the formula [49, §7.17] for ec (Γ \G(R)/K∞ , Vλ ), every term corresponding to M = G vanishes. Note that in [49] the right action of Hecke operators is considered: see [49, §7.19]. Thus to recover the trace of the usual left action of Hecke operators one has to exchange E (our Vλ ) and E ∗ in [49, §7.17]. Since we are only considering a unit Hecke operator, the orbital integrals at finite ∞ ) in [49, Theorem 7.14.B] vanish unless γ ∈ M(Q) is powerplaces denoted Oγ (fM bounded in M(Qp ) for every prime p. Recall that γ is also required to be elliptic in M(R), so that by the adelic product formula this condition on γ at all finite places implies that γ is also power-bounded in M(R). Thus it is enough to show that for any cuspidal Levi subgroup M of GR distinct from GR , and any powerbounded γ ∈ M(R), we have ΦM (γ , Vλ ) = 0 (where ΦM is defined on p. 498 of [49]). Choose an elliptic maximal torus T in MR such that γ ∈ T (R). Since the character of Vλ is already a continuous function on M(R), it is enough to show that there is a root α of T in G not in M such that α(γ ) = 1. All roots of T in M are imaginary, so it is enough to show that there exists a real root α of T in G such that α(γ ) = 1. Since γ is power-bounded in M(R), for any real root of T in G we have α(γ ) ∈ {±1}. Thus it is enough to show that there is a real root α such that α(γ ) > 0. This follows (for M = GR ) from the argument at the bottom of p. 499 in [49] (this is were the assumption that G is simply connected is used).  ) The proposition is a generalization of [60] to the orbifold case (Γ not neat) with non-trivial coefficients, but note that Harder’s formula is used in [49]. In particular for any g ≥ 1 and dominant weight λ we simply have Tell (Sp2g , λ) = ec (Ag , Vλ ) = e(Ag , Vλ ) by Poincaré duality # and self-duality of Vλ . As a special case we have the simple formula e(Ag ) = c∈C(Sp2g ) mc and Proposition 3 follows (with the help of a computer). See [100] for tables of masses mc , where the source code computing these masses (using [96]) can also be found. In the following table we record the value of e(Ag ) for small g. g e(Ag )

1 1

2 2

3 4 5 9

5 6 7 8 9 18 46 104 200 528

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The Euler characteristic of L2 -cohomology can also be evaluated explicitly. Theorem 3.3.4 in [98] expresses e(2)(Ag , Vλ ) in a (relatively) simple manner from Tell (Sp2g , λ ) for g ≤ g and dominant weights λ . Hence e(2)(Ag , Vλ ) can be derived from tables of masses, for any λ. Of course this does not directly yield dimensions of individual cohomology groups. Fortunately, Arthur’s endoscopic classification of automorphic representations for Sp2g allows us to write this Euler characteristic as a sum of two contributions: “old” contributions coming from automorphic representations for groups of lower dimension, and new contributions which only contribute to middle degree. Thus it is natural to try to compute old contributions by induction. As we shall see below, they can be described combinatorially from certain self-dual level one automorphic cuspidal representations of general linear groups over Q.

Arthur’s Endoscopic Classification in Level One • (A , V ) that can be deduced from We will explain the decomposition of H(2) g λ Arthur’s endoscopic classification. These real graded vector spaces are naturally endowed with a real Hodge structure, a Lefschetz operator and a compatible action of a commutative Hecke algebra. This last action will make the decomposition canonical. For this reason we will recall what is known about this Hecke action. We denote A for the adele ring of Q.

Definition 3 Let G be a reductive group over Z (in the sense of [92, Exposé XIX, Définition 2.7]). (i) If p is a prime, let Hpunr (G) be the commutative convolution algebra (“Hecke algebra”) of functions G(Zp )\G(Qp )/G(Zp ) → Q having finite support. (ii) Let Hfunr (G) be the commutative algebra of functions G( Z)\G(A)/G( Z) → Q having finite support, so that Hfunr (G) is the restricted tensor product :

unr p Hp (G). of a reductive group G, which we consider Recall the Langlands dual group G as a split reductive group over Q. We will mainly consider the following cases. G G

GLN

Sp2g

SO4g

SO2g+1

GLN

SO2g+1

SO4g

Sp2g

Recall from [53, 91] the Satake isomorphism: for F an algebraically closed field of characteristic zero and G a reductive group over Z, if we choose a square root

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of p in F then Q-algebra morphisms Hpunr (G) → F correspond naturally and ). bijectively to semisimple conjugacy classes in G(F (A), then it admits a If π is an automorphic cuspidal representation of GLN: decomposition as a restricted tensor product π = π∞ ⊗ p πp , where the last restricted tensor product is over all prime numbers p. Assume moreover that π GL (Z ) has level one, i.e. that πp N p = 0 for any prime p. Then π has the following invariants: (i) the infinitesimal character ic(π∞ ), which is a semisimple conjugacy class in glN (C) = MN (C) obtained using the Harish-Chandra isomorphism [62], (ii) for each prime number p, the Satake parameter c(πp ) of the unramified representation πp of GLN (Qp ), which is a semisimple conjugacy class in GLN (C) corresponding to the character by which Hpunr (GLN ) acts on the GLN (Zp )

complex line πp

.

We now introduce three families of automorphic cuspidal representations for general linear groups that will be exactly those contributing to intersection cohomology of Ag ’s. Definition 4 (i) For g ≥ 0 and integers w1 > · · · > wg > 0, let Oo (w1 , . . . , wg ) be the set of self-dual level one automorphic cuspidal representations π = π∞ ⊗ πf of GL2g+1(A) such that ic(π∞ ) has eigenvalues w1 > · · · > wg > 0 > −wg > π = SO2g+1 (C). · · · > −w1 . For π ∈ Oo (w1 , . . . , wg ) we let G (ii) For g ≥ 1 and integers w1 > · · · > w2g > 0, let Oe (w1 , . . . , w2g ) be the set of self-dual level one automorphic cuspidal representations π = π∞ ⊗ πf of GL4g (A) such that ic(π∞ ) has eigenvalues w1 > · · · > w2g > −w2g > · · · > π = SO4g (C). −w1 . For π ∈ Oe (w1 , . . . , w2g ) we let G (iii) For g ≥ 1 and w1 > · · · > wg > 0 with wi ∈ 1/2 + Z, let S(w1 , . . . , wg ) be the set of self-dual level one automorphic cuspidal representations π = π∞ ⊗ πf of GL2g (A) such that ic(π∞ ) has eigenvalues w1 > · · · > wg > π = Sp2g (C). −wg > · · · > −w1 . For π ∈ S(w1 , . . . , wg ) we let G These sets are all finite by [63, Theorem 1], and Oo (resp. Oe , S) is short for “odd orthogonal” (resp. “even orthogonal”, “symplectic”). A fact related to vanishing of cohomology with coefficients in Vλ for w(λ) odd is that Oo (w1 , . . . , wg ) = ∅ if w1 +· · ·+wg = g(g+1)/2 mod 2 and Oe (w1 , . . . , w2g ) = ∅ if w1 +· · ·+w2g = g mod 2. See [98, Remark 4.1.6] or [23, Proposition 1.8]. Remark 4 For small g the sets in Definition 4 are completely described in terms of level one (elliptic) eigenforms, due to accidental isomorphisms between classical groups in small rank (see [98, §6] for details). (i) For k > 0 the set S(k − 12 ) is naturally in bijection with the set of normalized eigenforms of weight 2k for SL2 (Z), and Oo (2k − 1) ' S(k − 12 ).

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2 (ii) For integers w1 > w2 > 0 such that w1 +w2 is odd, Oe (w1 , w2 ) ' S( w1 +w 2 )× 2 S( w1 −w 2 ).

Let us recall some properties of the representations appearing in Definition 4. Let Qreal be the maximal totally real algebraic extension of Q in C. Then Qreal is √ an infinite Galois extension of Q which contains p > 0 for any prime p. Theorem 28 For π as in Definition 4 there exists a finite subextension E of Qreal /Q such that for any prime number p, the characteristic polynomial of pw1 c(πp ) has coefficients in E. Moreover c(πp ) is compact (i.e. power-bounded). Proof That there exists a finite extension E of Q satisfying this condition is a special case of [24, Théorème 3.13]. The fact that it can be taken totally real follows from unitarity and self-duality of π. The last statement is a consequence of [95] and [25].  ) Let E(π) be the smallest such extension. Then πf is defined over E(π), and this structure is unique up to C× /E(π)× . There is an action of the Galois group Gal(Qreal /Q) on Oo (w1 , . . . ) (resp. Oe (w1 , . . . ), S(w1 , . . . )): if π = π∞ ⊗ πf , σ (π) := π∞ ⊗ σ (πf ) belongs to the same set. Dually we have c(σ (π)p ) = p−w1 σ (pw1 c(πp )). For π ∈ Oo (w1 , . . . ) or Oe (w1 , . . . ) the power of p is not necessary since w1 ∈ Z. π (C), In all three cases c(πp ) can be lifted to a semisimple conjugacy class in G uniquely except in the second case, where it is unique only up to conjugation in O4g (C). Identifying semisimple conjugacy classes to Weyl group orbits in maximal tori, this is elementary. We abusively denote this conjugacy class by c(πp ). We now consider the Archimedean place of Q, which will be of particular importance for real Hodge structures. Recall that the Weil group of R is defined as the non-trivial extension of Gal(C/R) by C× . If H is a complex reductive group and ϕ : C× → H (C) is a continuous semisimple morphism, there is a maximal torus T of H such that ϕ factors through T (C) and takes the form z → zτ1 z¯ τ2 for uniquely determined τ1 , τ2 ∈ X∗ (T ) ⊗Z C such that τ1 − τ2 ∈ X∗ (T ). Here X∗ (T ) is the group of cocharacters of T and zτ1 z¯ τ2 is defined as (z/|z|)τ1 −τ2 |z|τ1 +τ2 . We call the H (C)-conjugacy class of τ1 in h = Lie(H ) (complex analytic Lie algebra) the infinitesimal character of ϕ, denoted ic(ϕ). If ϕ : WR → H (C) is continuous semisimple we let ic(ϕ) = ic(ϕ|C× ). In all three cases in Definition 4 there is a continuous semisimple morphism π (C) such that for any z ∈ C× , ϕπ∞ (z) has eigenvalues ϕπ∞ : WR → G ⎧ ±w1 , . . . , (z/¯ ⎪ z)±wg , 1 ⎪ ⎨(z/¯z) (z/¯z)±w1 , . . . , (z/¯z)±w2g

⎪ ⎪ ⎩(z/|z|)±2w1 , . . . , (z/|z|)±2wg

if π ∈ Oo (w1 , . . . , wg ), if π ∈ Oe (w1 , . . . , w2g ), if π ∈ S(w1 , . . . , wg ),

π . The parameter ϕπ∞ is characterized up to in the standard representation of G conjugacy by this property except in the second case where it is only characterized

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up to O4g (C)-conjugacy. To rigidify the situation we choose a semisimple conjugacy π whose image via the standard representation has class τπ in the Lie algebra of G eigenvalues ±w1 , . . . , ±wg , 0 (resp. ±w1 , . . . , ±w2g , resp. ±w1 , . . . , ±wg ). Then π , τπ ) is well-defined up to isomorphism unique up to conjugation by the pair (G  Gπ , since in the even orthogonal case τπ is not fixed by an outer automorphism of π . Up to conjugation by G π there is a unique ϕπ∞ : WR → G π (C) as above and G π (C) is finite. such that ic(ϕπ∞ ) = τπ . In all cases the centralizer of ϕπ∞ (WR ) in G Let us now indicate how the general definition of substitutes for ArthurLanglands parameters in [9] specializes to the case at hand. Definition 5 Let Vλ be an irreducible algebraic representation of Sp2g , given by the dominant weight λ = (λ1 ≥ · · · ≥ λg ≥ 0). Let ρ be half the sum of the positive roots for Sp2g , and τ = λ + ρ = (w1 > · · · > wg > 0) where wi = λi + n + 1 − i ∈ Z, which we can see as the regular semisimple conjugacy class in so2g+1(C). Let unr,τ Ψdisc (Sp2g ) be the set of pairs (ψ0 , {ψ1 , . . . , ψr }) with r ≥ 0 and such that (0)

(0)

(i) ψ0 = (π0 , d0 ) where π0 ∈ Oo (w1 , . . . , wg0 ) and d0 ≥ 1 is an odd integer, (ii) The ψi ’s, for i ∈ {1, . . . , r}, are distinct pairs (πi , di ) where di ≥ 1 is an inte(i) (i) (i) (i) ger and πi ∈ Oe (w1 , . . . , wgi ) with gi even (resp. πi ∈ S(w1 , . . . , wgi )) if di is odd (resp. even). # (iii) 2g + 1 = (2g0 + 1)d0 + ri=1 2gi di , (iv) The sets a. { d02−1 , d02−3 , . . . , 1}, d0 −1 b. {w1(0) + d02−1 − j, . . . , wg(0) 0 + 2 − j } for j ∈ {0, . . . , d0 − 1}, (i) di −1 (i) di −1 c. {w1 + 2 −j, . . . , wgi + 2 −j } for i ∈ {1, . . . , r} and j ∈ {0, . . . , di − 1} are disjoint and their union equals {w1 , . . . , wg }. We will write more simply ψ = ψ0 · · · ψr = π0 [d0 ] · · · πr [dr ]. This could be defined as an “isobaric sum”, but for the purpose of this appendix we can simply consider this expression as a formal unordered sum. If π0 is the trivial representation of GL1 (A), we write [d0 ] for 1[d0], and when d = 1 we simply write π for π[1]. Example 1 unr,ρ

(i) For any g ≥ 1, [2g + 1] ∈ Ψdisc (Sp2g ). unr,ρ (ii) [9] Δ11 [2] ∈ Ψdisc (Sp12 ) where Δ11 ∈ S( 11 2 ) is the automorphic representation of GL2 (A) corresponding to the Ramanujan Δ function. (iii) For g ≥ 1, τ = (w1 > · · · > wg > 0), for any π ∈ Oo (w1 , . . . , wg ) we have unr,τ π = π[1] ∈ Ψdisc (Sp2g ).

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Definition 6

$ unr,τ  (Sp2g ), let Lψ = ri=0 G (i) For ψ = π0 [d0 ] · · · πr [dr ] ∈ Ψdisc πi (C). ˙ Let ψ : Lψ × SL2 (C) → SO2n+1 (C) be a morphism such that composing with the standard representation gives 0≤i≤r StdG ⊗νdi where StdG is  πi πi  the standard representation of Gπi and νdi is the irreducible representation of SL2 (C) of dimension di . Then ψ˙ is well-defined up to conjugation by SO2n+1 (C). (ii) Let Sψ be the centralizer of ψ˙ in SO2g+1(C), which is isomorphic to (Z/2Z)r . A basis is given by (si )1≤i≤r where si is the image by ψ˙ of the non-trivial  element in the center of G πi . (iii) Let ψ∞ be the morphism WR ×SL2 (C) → SO2g+1 (C) obtained by composing  ψ˙ with the morphisms ϕπi,∞ : WR → G of ψ∞ in πi (C). The centralizer Sψ∞# SO2g+1 (C) contains Sψ and is isomorphic to (Z/2Z)x where x = ri=1 gi . (iv) For p a prime let cp (ψ) be the image of ((c(πi,p ))0≤i≤r , diag(p1/2 , p−1/2 )) ˙ a well-defined semisimple conjugacy class in SO2g+1 (C). Let χp (ψ) : by ψ, unr Hp (Sp2g ) → C$be the associated character. unr (v) Let χf (ψ) = p χp (ψ) : Hf (Sp2g ) → C be the product of the χp (ψ)’s. It takes values in the smallest subextension E(ψ) of Qreal containing E(π0 ), . . . , E(πr ), which is also finite over Q. For any prime p the characteristic polynomial of cp (ψ) has coefficients in E(ψ). In particular we have a unr,τ continuous action of Gal(Qreal /Q) on Ψdisc (Sp2g ) which is compatible with χf . The following theorem is a consequence of [72].

unr,τ Theorem 29 For any g ≥ 1 the map (λ, ψ ∈ Ψdisc (Sp2g )) → χf (ψ) is injective.

The last condition in Definition 5 is explained by compatibility with infinitesimal characters, stated after the following definition. Definition 7 Let δ∞ : C× −→ C× × SL2 (C) z −→ (z, diag(||z||1/2, ||z||−1/2)). For a complex reductive group H and a morphism ψ∞ : C× × SL2 (C) → H (C) which is continuous semisimple and algebraic on SL2 (C), let ic(ψ∞ ) = ic(ψ∞ ◦ δ∞ ). Similarly, if ψ∞ : WR × SL2 (C) → H (C), let ic(ψ∞ ) = ic(ψ∞ |C× ). unr,τ (Sp2g ), we have ic(ψ∞ ) = τ (equality between semisimple For ψ ∈ Ψdisc conjugacy classes in so2g+1 (C)), and this explains the last condition in Definition 5. For ψ as above Arthur constructed [9, Theorem 1.5.1] a finite set Π(ψ∞ ) of irreducible unitary representations of Sp2g (R) and a map Π(ψ∞ ) → Sψ∨∞ , where A∨ = Hom(A, C× ). We simply denote this map by π∞ → *·, π∞ ,. Arthur also defined a character ψ of Sψ , in terms of symplectic root numbers. We do not recall the definition, but note that for everywhere unramified parameters considered

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here, this character can be computed easily from the infinitesimal characters of the πi ’s (see [23, §3.9]). We can now formulate the specialization of [9, Theorem 1.5.2] to level one and algebraic regular infinitesimal character, and its consequence for L2 -cohomology thanks to [17]. Theorem 30 Let g ≥ 1. Let Vλ be an irreducible algebraic representation of Sp2g with dominant weight λ. Let τ = λ + ρ. The part of the discrete automorphic spectrum for Sp2g having level Sp2g ( Z) and infinitesimal character τ decomposes as a completed orthogonal direct sum 

L2disc (Sp2g (Q)\ Sp2g (A)/ Sp2g ( Z))ic=τ '



π∞ ⊗ χf (ψ).

unr,τ ψ∈Ψdisc (Sp2g ) π∞ ∈Π(ψ∞ ) *·,π∞ ,=ψ

Therefore • H(2) (Ag , Vλ ) '





H • ((g, K), π∞ ⊗ Vλ ) ⊗ χf (ψ)

unr,τ ψ∈Ψdisc (Sp2g ) π∞ ∈Π(ψ∞ ) *·,π∞ ,=ψ

where as before g = sp2g (C) and K = U (g). To be more precise the specialization of Arthur’s theorem relies on [98, Lemma 4.1.1] and its generalization giving the Satake parameters, considering traces of arbitrary elements of the unramified Hecke algebra. unr,τ (Sp2g ) the sets Π(ψ∞ ) and characters *·, π∞ , for Thanks to [4] for ψ ∈ Ψdisc π∞ ∈ Π(ψ∞ ) are known to coincide with those constructed by Adams and Johnson in [1]. Furthermore, the cohomology groups H • ((g, K), π∞ ⊗ Vλ )

(33)

for π∞ ∈ Π(ψ∞ ) were computed explicitly in [102, Proposition 6.19], including the real Hodge structure. Thus in principle one can compute (algorithmically) the dimensions of the i (A , V ) if the cardinalities of the sets O (. . . ), O (. . . ) cohomology groups H(2) g λ o e and S(. . . ) are known. As explained in the previous section, for small g the Euler characteristic e(2) (Ag , Vλ ) can be evaluated using the trace formula. For unr,τ π ∈ Oo (τ ) ⊂ Ψdisc (Sp2g ), the contribution of π to the Euler characteristic expanded using Theorem 30 is simply (−1)g(g+1)/22g = 0. The contributions of unr,τ other elements of Ψdisc (Sp2g ) to e(2) (Ag , Vλ ) can be evaluated inductively, using the trace formula and the analogue of Theorem 30 also for the groups Sp2g for g < g, SO4m for m ≤ g/2 and SO2m+1 for m ≤ g/2. We refer to [98] for details, and simply emphasize that computing the contribution of some ψ to the Euler characteristic is much easier than computing all dimensions using [102]. For

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unr,τ example for ψ ∈ Ψdisc (Sp2g ) we have



e((sp2g , K), π∞ ⊗ Vλ ) = ±2g−r

(34)

π∞ ∈Π(ψ∞ ) *·,π∞ ,|Sψ =ψ

where the integer r is as in Definition 5 and the sign is more subtle but easily computable. To sum up, using the trace formula, we obtained tables of cardinalities for the three families of sets in Definition 4. See [100]. For small λ, more precisely for g + λ1 ≤ 11, there is another way to enumerate unr,τ all elements of Ψdisc (Sp2g ), which still relies on some computer calculations, but much simpler ones and of a very different nature. The following is a consequence of [22, Théorème 3.3]. The proof uses the Riemann-Weil explicit formula for automorphic L-functions, and follows work of Stark, Odlyzko and Serre for zeta functions of number fields giving lower bound of their discriminants, and of Mestre, Fermigier and Miller for L-functions of automorphic representations. The striking contribution of [22, Théorème 3.3] is the fact that the rank of the general linear group is not bounded a priori, but for the purpose of the present appendix we impose regular infinitesimal characters. Theorem 31 For w1 ≤ 11, the only non-empty Oo (w1 , . . . ), Oe (w1 , . . . ) or S(w1 , . . . ) are the following. (i) For g = 0, Oo () = {1}. (ii) For 2w1 ∈ {11, 15, 17, 19, 21}, S(w1 ) = {Δ2w1 } where Δ2w1 corresponds to the unique eigenform in S2w1 +1 (SL2 (C)(Z)). (iii) Oo (11) = {Sym2 (Δ11 )} (Sym2 functoriality was constructed in [44]), (iv) For (2w1 , 2w2 ) ∈ {(19, 7), (21, 5), (21, 9), (21, 13)}, S(w1 , w2 ) = {Δ2w1 ,2w2 }. These correspond to certain Siegel eigenforms in genus two and level one. Of course this is coherent with our tables. As a result, for g + λ1 ≤ 11, i.e in 11 

card {λ | w(λ) even and g + λ1 ≤ 11} = 1055

g=1 unr,τ non-trivial cases, Ψdisc (Sp2g ) can be described explicitly in terms of the 11 automorphic representations of general linear groups appearing in Theorem 31. In unr,τ most of these cases Ψdisc (Sp2g ) is just empty, in fact

 g+λ1 ≤11

with 146 non-vanishing terms.

unr,τ card Ψdisc (Sp2g ) = 197

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Corollary 3 For g = 3 and λ = (1, 1, 0) we have I H • (A3Sat , Vλ ) = 0. unr,τ (Sp6 ) = ∅. Proof Using Theorem 31 and Definition 5 we see that Ψdisc

 )

Remark 5 Of course this result also follows from [98]. More precisely, without the unr,τ a priori knowledge given by Theorem 31 we have for λ = (1, 1, 0) that Ψdisc (Sp6 ) is the disjoint union of Oo (4, 3, 1) with the two sets / 0 π1 π2 | π1 ∈ Oo (w1 ), π2 ∈ Oe (w1 , w2 ) with {w1 , w1 , w2 } = {4, 3, 1} , {π1 π2 [2] | π1 ∈ Oo (1), π2 ∈ S(7/2)} . By Remark 4 and vanishing of S2k (SL2 (Z)) for 0 < k < 6 both sets are empty, so Corollary 3 follows from the computation of e(2) (A3 , Vλ ) = 0. Note that computationally this result is easier than Proposition 3, since computing masses for Sp8 is much more work than for Sp6 . If we now focus on λ = 0, we have the following classification result. unr,ρ

Corollary 4 For 1 ≤ g ≤ 5 we have Ψdisc (Sp2g ) = {[2g + 1]}. For 6 ≤ g ≤ 11, unr,ρ all elements of Ψdisc (Sp2g )  {[2g + 1]} are listed in the following tables. g 6 7 11

unr,ρ

Ψdisc (Sp2g )  {[2g + 1]} Δ11 [2] [9] Δ11 [4] [7] Δ21 [2] [19] Δ21 [2] Δ11 [8] [3] Δ21,5 [2] Δ17 [2] Δ11 [4] [3] Δ21 [2] Δ17 [2] Δ11 [4] [7] Δ19 [4] Δ11 [4] [7] Δ21,9 [2] Δ15 [4] [7] Δ15 [8] [7] Δ21 [2] Δ17 [2] [15] Δ19 [4] [15] Δ11 [10] Sym2 (Δ11 ) Δ17 [6] [11] Δ21 [2] Δ15 [4] [11] Δ21,13[2] Δ17 [2] [11]

unr,ρ

g Ψdisc (Sp2g )  {[2g + 1]} 8 Δ11 [6] [5] Δ15 [2] Δ11 [2] [9] Δ15 [2] [13] 9 Δ11 [8] [3] Δ17 [2] Δ11 [4] [7] Δ17 [2] [15] Δ15 [4] [11] 10 Δ19,7[2] Δ15 [2] Δ11 [2] [5] Δ19 [2] Δ11 [6] [5] Δ11 [10] [1] Δ19 [2] [17] Δ19 [2] Δ15 [2] Δ11 [2] [9] Δ15 [6] [9] Δ17 [4] Δ11 [2] [9] Δ19 [2] Δ15 [2] [13] Δ17 [4] [13]

In the next section we will recall and make explicit the description by Langlands and Arthur of L2 -cohomology in terms of ψ∞ , which is simpler than using [102]

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directly but imposes to work with the group GSp2g (which occurs in a Shimura datum) instead of Sp2g . unr,ρ Let us work out the simple case of ψ = [2g + 1] ∈ Ψdisc (Sp2g ) directly using [102, Proposition 6.19], using their notation, in particular g = k ⊕ p (complexification of the Cartan decomposition of g0 = Lie(Sp2g (R)) and p = p+ ⊕ p− (decomposition of p according to the action of the center of K, which is isomorphic to U (1)). In this case Π(ψ∞ ) = {1}, u = 0 and l = sp2g , so H 2k ((g, K), 1) = H k,k ((g, K), 1) ' HomK

! 2k .

" p, C

and the dimension 1 of this space is the number of constituents in the multiplicityfree representation k (p+ ). One can show (as for any hermitian symmetric space) that this number equals the number of elements of length p in W (Sp2g )/W (K) ' W (Sp2g )/W (GLg ). A simple explicit computation that we omit shows that this number equals the number of partitions of k as a sum of distinct integers between 1 and n. In conclusion, 

T i dim H i ((sp2n , K), 1) =

i

n 

(1 + T 2k ).

k=1

Combining the first part of Corollary 4 and this computation we obtain Theorem 17. Theorem 32 For 1 ≤ g ≤ 5 we have I H • (AgSat , Q) ' Rg as graded vector spaces over Q. We can sharpen Theorem 25 in the particular case of level one, although what we obtain is not a stabilization result (see the remark after the theorem). Theorem 33 For g ≥ 2, λ a dominant weight for Sp2g and k < 2g − 2, IH

k

(AgSat , Vλ )

=

2 Rgk 0

if λ = 0 otherwise

where Rgk denotes the degree k part of Rg , ui having degree 2i. unr,τ (Sp2g ) different from [2g + 1], one sees easily from the Proof For ψ ∈ Ψdisc construction in [1] that the trivial representation of Sp2g (R) does not belong to Π(ψ∞ ). So using Zucker’s conjecture, Borel-Casselman and Arthur’s multiplicity unr,τ formula, we are left to show that for ψ ∈ Ψdisc (Sp2g )  {[2g + 1]}, for any • π∞ ∈ Π(ψ∞ ), H ((g, K), π∞ ⊗ Vλ ) vanishes in degree less than 2g − 2. We can read this from [102, Proposition 6.19], and we use the notation from this paper. Let θ be the Cartan involution of Sp2g (R) corresponding to K, so that p = g−θ . The representation π∞ is constructed from a θ -stable parabolic subalgebra q = l ⊕ u of g, where l is also θ -stable. We will show that dim u ∩ p ≥ 2g − 2. The (complex)

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$ # Lie algebra l is isomorphic to j gl(aj + bj ) × sp2c with c + j aj + bj = g. The action of the involution θ on the factor gl(ak + bk ) is such that the associated real Lie algebra is isomorphic to u(ak , bk ). Using notation of Definition 5, the integer c equals (d0 −1)/2. Since ψ = [2g +1] we have r ≥ 1 and this implies that c ≤ g −2 (this is particular to level one, in arbitrary level one would simply get c < g). We have 2 dim u ∩ p + dim l ∩ p = dim p = g(g + 1) since l, u and its opposite Lie algebra with respect to l are all stable under θ . We compute dim l ∩ k =

 (aj2 + bj2 ) + c2 j

and so dim l ∩ p = dim l − dim l ∩ k = 2



aj bj + c(c + 1).

j

We get dim u ∩ p =

g(g + 1) c(c + 1)  aj bj . − − 2 2 j

We have 

aj bj ≤ (

j

 j

aj )(

 j

bj ) ≤

(g − c)2 4

which implies dim u ∩ p ≥

g(g + 1) c(c + 1) (g − c)2 − − . 2 2 4

The right hand side is a concave function of c, thus its maximal value for c ∈ {0, . . . , g − 2} is g(g + 1)/2 − min(g 2 /4, (g − 2)(g − 1)/2 + 1). For integral g = 3 one easily checks that this equals g(g + 1)/2 − (g − 2)(g − 1)/2 − 1 = 2g − 2. For g = 3 we get 2g − 2 − 1/4, and $2g − 2 − 1/4% = 2g − 2.  ) Remark 6 (i) For k ≥ g even the surjective map Q[λ1 , λ3 , . . . ]k → Rgk has non-trivial kernel, so in the range g ≤ k < 2g − 2 we do not get stabilization. (ii) One can show that for the trivial system of coefficients (i.e. λ = 0) the bound in Theorem 33 is sharp for even g ≥ 6, is not sharp for odd

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g (i.e. I H 2g−2 (AgSat , Q) = Rg ) but that for odd g ≥ 9 we have I H 2g−1 (AgSat , Q) = 0. For even g ≥ 6 and odd g ≥ 9 this is due to ψ of the form π[2] [2g − 3] where π ∈ S(g − 1/2) corresponds to a weight 2g eigenform for SL2 (Z). (iii) Of course for non-trivial λ one can improve on this result, e.g. for λ1 > · · · > λg > 0 we have vanishing in degree = g(g + 1)/2 (see [88, Theorem 5] and [75, Theorem 5.5] for a vanishing result for ordinary cohomology). If we only assume λg > 0, this forces c = 0 in the proof and we obtain vanishing in degree less than (g 2 + 2g)/4. (iv) An argument similar to the proof of Theorem 33 can be used to show the same result for k < g in arbitrary level. It seems likely that one could extend the sharper bound in Theorem 33 to certain deeper levels, e.g. Iwahori level at a finite number of primes. There is also a striking consequence of [102, Proposition 6.19] (and [74, Lemma 9.1]) that was observed in [80], namely the fact that any ψ only contributes in degrees of a certain parity. This implies the dimension part in the following proposition which is a natural first step towards the complete description in the next section. Proposition 5 There is a canonical decomposition 

Qreal ⊗Q I H • (AgSat , Vλ ) =

Qreal ⊗E(ψ) Hψ•

unr,τ ψ∈Ψdisc (Sp2g )

where Hψ• is a graded vector space of total dimension 2n−r (r as in Definition 5) over the totally real number field E(ψ), endowed with (i) for any n ≥ 0, a pure Hodge structure of weight n on Hψn , inducing a bigrading  p,q C ⊗E(ψ) Hψn = p+q=n Hψ , (ii) a linear operator L : R ⊗E(ψ) Hψ• → R ⊗E(ψ) Hψ• mapping Hψ and such that for any 0 < n ≤ g(g + 1)/2,

p,q

g(g+1)/2−n

Ln : R ⊗E(ψ) Hψ

p+1,q+1

to Hψ

g(g+1)/2+n

→ R ⊗E(ψ) Hψ

is an isomorphism. This decomposition is Hfunr (Sp2n )-equivariant, the action on Hψ being by the character χf (ψ). Proof Recall that by Zucker’s conjecture [76, 78, 89] we have a Hecke-equivariant isomorphism • I H • (AgSat , Vλ ) ⊗Q R ' H(2) (Ag , Vλ ⊗Q R).

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By Theorem 30 there are graded vector spaces Hψ that can be defined over Eψ such that the left hand side is isomorphic to the right hand side. By Theorem 29 each summand on the right hand side can be cut out using Hecke operators, so the decomposition is canonical and the E(ψ)-structure on Hψ is canonical as well. We • (A , R ⊗ V ) with the real Hodge structure endow R ⊗Q I H • (AgSat , Vλ ) ' H(2) g Q λ given by Hodge theory on L2 -cohomology of the non-compact Kähler manifold Ag . There is a natural Lefschetz operator L given by cup-product with the Kähler form. It commutes with Hecke operators and one can check that L is i times the operator X defined on p. 60 of [7]. The hard Lefschetz property of L is known both in L2 cohomology and (g, K)-cohomology. It follows from [80, Theorem 1.5] that any ψ contributes in only one parity, so the claim about dimE(ψ) Hψ follows from (34).  )  unr,τ If o is any Gal(Qreal /Q)-orbit in Ψdisc (Sp2g ), ψ∈o Hψ is naturally defined over Q and endowed with an action of a quotient of Hfunr (Sp2g ) which is a finite totally real field extension E(o) of Q, and elements of o correspond bijectively to Q-embeddings E(o) → Qreal . Remark 7 There is also a rational Hodge structure defined on intersection cohomology groups thanks to Morihiko Saito’s theory of mixed Hodge modules, but unfortunately it is not known whether the induced real Hodge structure coincides with the one defined using L2 theory (see [64, §5]). Similarly, there is another natural Lefschetz operator acting on the cohomology I H • (AgSat , Vλ ) (using the first Chern class of an ample line bundle on AgSat ), and it does not seem obvious that it coincides (up to a real scalar) with the Kähler operator L above, although it could perhaps be deduced from arguments as in [47, §16.6].

Description in Terms of Archimedean Arthur-Langlands Parameters Langlands and Arthur [7, 8] gave a conceptually simpler point of view on the Hodge structure with Lefschetz operator on L2 -cohomology. This applies to Shimura varieties and so one would have to work with the reductive group GSp2g instead of Sp2g , since only GSp2g is part of a Shimura datum, (GSp2g , Hg  Hg ). Very roughly, the idea of this description for a Shimura datum (G, X) is that for K1 the stabilizer in G(R) of a point in X, representations of G(R) in an Adams-Johnson packet are parametrized by certain cosets in W (G, T )/W (K1 , T ) for a maximal torus T of G(R) contained in K1 , and W (K1 , T ) is also identified with the stabilizer of the cocharacter μ : GL1 (C) → G(C) obtained from the Shimura datum. This cocharacter can be seen as an extremal weight for an irreducible algebraic and rμ is minuscule, i.e. its weights form a single orbit under representation rμ of G the Weyl group of G, which is identified with W (G, T ).

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 In the case of GSp2g the Langlands dual group is GSp 2g = GSpin2g+1 and rμ is a spin representation. Morphisms taking values in a spin group cannot simply be described as self-dual linear representations. For this reason we do not have substitutes for Arthur-Langlands parameters for GSp2g constructed using automorphic representations of general linear groups (that is the analogue of 5 for GSp2g and arbitrary level), and no precise multiplicity formula yet. Bin Xu [103] obtained a multiplicity formula in many cases, but his work does not cover the case of non-tempered Arthur-Langlands parameters that is typical when λ = 0. For example all parameters appearing in Corollary 4 are non-tempered. Fortunately in level one it turns out that we can simply formulate the result in terms of Sp2g . This is in part due to the fact that, letting K1 = R>0 Sp2g (R) ⊂ GSp2g (R), the natural map Ag = Sp2g (Q)\ Sp2g (A)/K Sp2g ( Z) → GSp2g (Q)\ GSp2g (A)/K1 GSp2g ( Z) (35) is an isomorphism. This is a special case of a more general principle in level one, see §4.3 and Appendix B in [23] for a conceptual explanation. Since the cohomology of the intermediate extension to AgSat of Vλ vanishes when the weight w(λ) := λ1 + · · · + λg is odd, we will be able to formulate the result using Spin2g+1 (C) instead of GSpin2g+1 (C). Let g ≥ 1 and λ a dominant weight for Sp2g , as usual let τ = λ + ρ. Consider unr,τ ψ = ψ0 · · · ψr = π0 [d0 ] · · · πr [dr ] ∈ Ψdisc (Sp2g )

as in Definition 5. First we recall how to equip C ⊗E(ψ) Hψ• with a continuous semisimple linear action ρψ of C× × SL2 (C). This action will be trivial on R>0 ⊂ C× by construction. (i) We let z ∈ C× act on Hψ by multiplication by (z/|z|)q−p . (ii) There is a unique algebraic action of SL2 (C) on C ⊗E(ψ) Hψ• such that the % & 01 action of ∈ sl2 is given by the Lefschetz operator L and the diagonal 00 torus in SL2 (C) preserves the grading on C ⊗E(ψ) Hψ• . Explicitly, by hard Lefschetz we have that for t ∈ GL1 , diag(t, t −1 ) ∈ SL2 acts on Hψi by multiplication by t g(g+1)/2−i . This algebraic action is defined over R, and if we knew that L is rational it would even be defined over E(ψ). (iii) These actions commute and we obtain p,q

ρψ : C× × SL2 (C) −→ GL(C ⊗E(ψ) Hψ ). The dimension g(g + 1)/2 being fixed, we see that the isomorphism class of the real Hodge structure with Lefschetz operator (R ⊗E(ψ) Hψ , L) determines and is

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determined by the isomorphism class of ρψ . In fact they are both determined by the Hodge diamond of R ⊗E(ψ) Hψ• . To state the description of these isomorphism classes in terms of ArthurLanglands parameters we need a few more definitions. For i ∈ {0, . . . , r} let mi  be the product # of di with the dimension of the standard representation of G πi , so r that 2g + 1 = i=0 mi . Let Mψ0 = SOm0 (C). For 1 ≤ i ≤ r let (Mψi , τψi ) be a pair such that Mψi ' SOmi (C) and τψi is a semisimple element in the Lie algebra of Mψi whose image via the standard representation has eigenvalues & % & % di − 1 di − 1 − j , . . . , ± wg(i)i + − j for 0 ≤ j ≤ di − 1. ± w1(i) + 2 2 As in Definition 6 the point of this definition is that the group of automorphisms of (Mψi , τψi ) is the adjoint group of Mψi , because τψi is not invariant under the outer automorphism$ of Mψi . Note that Mψi is semisimple since mi = 2. Let Mψ = 0≤i≤r Mψi . There is a natural embedding ιψ : Mψ → SO2g+1 (C). Up to conjugation by Mψ there is a unique morphism fψ : Lψ × SL2 (C) → Mψ such that ˙ which implies that fψ is an algebraic morphism, (i) ιψ ◦ fψ is conjugated to ψ, (ii) the differential of fψ maps ((τπi )1≤i≤r , diag( 12 , − 12 )) to τψ := (τψi )0≤i≤r . ˙ so we assume this equality We can conjugate ψ˙ in SO2g+1 (C) so that ιψ ◦ fψ = ψ, from now on. The centralizer of ιψ in SO2g+1 (C) coincides with Sψ . Let   fψ,∞ = fψ ◦ (ϕπi,∞ )0≤i≤r , IdSL2 (C) : WR × SL2 (C) → Mψ . Condition (ii) above to ic(fψ,∞ ) = τψ . We have ψ∞ = ιψ ◦ fψ,∞ . $ is equivalent$ Let Mψ,sc = ri=0 Mψi ,sc ' ri=0 Spinmi (C) be the simply connected cover of Mψ . Let spinψ0 be the spin representation of Mψ0 ,sc , of dimension 2(m0 −1)/2 . The group Mψi ,sc has two half-spin representations spin± ψi , distinguished by the fact that + the largest eigenvalue of spinψi (τψi ) is greater than that of spin− ψi (τψi ). They both $  have dimension 2mi /2−1 . Let Lψ,sc = ri=0 (G ) be the simply connected cover πi sc of Lψ , a product of spin and symplectic groups. There is a unique algebraic lift f ψ : Lψ,sc ×SL2 (C) → Mψ,sc of fψ . There is a unique algebraic lift ι ψ : Mψ,sc → Spin2g+1 (C) of ιψ and it has finite kernel. The pullback of the spin representation of Spin2g+1 (C) via ι ψ decomposes as spinψ0 ⊗

 (i )i ∈{±}r

spinψ11 ⊗ · · · ⊗ spinψrr .

(36)

Let us be more specific. It turns out that the preimage Sψ,sc ' (Z/2Z)r+1 of Sψ in Spin2g+1 (C) commutes with ι ψ (Mψ,sc ). This is specific to conductor one, i.e. level Sp2g (Z). Thus the spin representation of Spin2g+1 (C) restricts to a representation

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of Mψ,sc × Sψ,sc , and (36) is realized by decomposing into isotypical components for Sψ,sc . More precisely, the non-trivial element of the center of Spin2g+1 (C) is mapped to −1 in the spin representation, and there is a natural basis (si )1≤i≤r of Sψ over (Z/2Z)r and lifts (˜si )1≤i≤r in Sψ,sc such that in each factor of (36) s˜i acts by i . ×  For 0 ≤ i ≤ r there is a unique continuous lift ϕ πi,∞ : C → (G πi )sc of the restriction ϕπi,∞ |C× . One could lift morphisms from WR but the lift is not unique in general. Finally we can define   ×  f  ψ,∞ = f ψ ◦ (ϕ πi,∞ )0≤i≤r , IdSL2 (C) : C × SL2 (C) → Mψ . Theorem 34 For g ≥ 1, λ a dominant weight for Sp2g and unr,τ (Sp2g ) ψ = ψ0 · · · ψr ∈ Ψdisc

we have an isomorphism of continuous semisimple representations of C× ×SL2 (C):   ρψ ' spinψ0 ⊗ spinuψ11 ⊗ · · · ⊗ spinuψrr ◦ f ψ,∞ where u1 , . . . , ur ∈ {+, −} can be determined explicitly (see [99] for details). Proof This is essentially a consequence of [7, Proposition 9.1] and Arthur’s multiplicity formula, but we need to argue that in level one the argument goes through with the multiplicity formula for Sp2g (Theorem 30) instead. This is due to two simple facts. Firstly, the group K1 in [7, §9] is simply R>0 × K, and this implies that for π∞ a unitary irreducible representation of GSp2g (R)/R>0 , H • ((gsp2g , K1 ), π∞ ⊗ Vλ ) = H • ((sp2g , K), π∞ |Sp2g (R) ⊗ Vλ ) where on the left hand side Vλ is seen as an algebraic representation of PGSp2g (since we can assume that w(λ) even). Secondly, as we observed above the preimage Sψ,sc of Sψ in Spin2g+1 (C) still commutes with ι ψ (Mψ,sc ), making the representation σψ of [7, §9] well-defined. This second fact is particular to the level one case.  ) unr,τ (Sp2g ), making the decomposition in ProposiTo conclude, if we know Ψdisc tion 5 completely explicit boils down to computing signs (ui )1≤i≤r and branching in the following cases:

(i) for the morphism Spin2a+1 × SL2 → Spin(2a+1)(2b+1) lifting the representation StdSO2a+1 ⊗ Sym2b (StdSL2 ) : SO2a+1 × SL2 → SO(2a+1)(2b+1) and the spin representation of Spin(2a+1)(2b+1),

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(ii) for Spin4a × SL2 → Spin4a(2b+1) and both half-spin representations, (iii) for Sp2a × SL2 → Spin4ab and both half-spin representations. For example one can using Corollary 4, Proposition 5 and Theorem 34 one can explicitly compute I H • (Ag ) for all g ≤ 11. Example 2 (i) For any g ≥ 1 and ψ = [2g + 1], the group Lψ is trivial and up to a shift we recover the graded vector space Rg as the composition of the spin representation of Spin2g+1 composed with the principal morphism SL2 → Spin2g+1 , graded by weights of a maximal torus of SL2 . (ii) Consider g = 6 and ψ = Δ11 [2] [9]. For ψ0 we have spinψ0 ◦f; ψ0 ' ν11 ⊕ ν5 , where as before νd denotes the irreducible d-dimensional representation of SL2 . For ψ1 = Δ11 [2] we have Lψ1 = Sp2 (C), Mψ1 = SO4 (C), − ; ; spin+ ψ1 ◦fψ1 ' StdSp2 ⊗1SL2 and spinψ1 ◦fψ1 ' 1Sp2 ⊗ ν2 . It turns out that u1 = −, so ρψ |C× is trivial and ρψ |SL2 ' (ν11 ⊕ ν5 ) ⊗ ν2 ' ν12 ⊕ ν10 ⊕ ν6 ⊕ ν4 . Thus Hψ• has primitive cohomology classes in degrees 10, 12, 16, 18 (a factor νd contributes a primitive cohomology class in degree g(g + 1)/2 − d + 1). Surprisingly, these classes are all Hodge, i.e. they belong to Hψ2k ∩Hψk,k , despite the fact that the parameter ψ is explained by a non-trivial motive over Q (attached to Δ11 ). (iii) Consider g = 7 and ψ = Δ11 [4] [7]. Again Lψ,sc ' Sp2 (C). For ψ0 = [7] we have spinψ0 ◦f; ψ0 ' ν7 ⊕ 1. For ψ1 = Δ11 [4] we have Lψ1 = Sp2 (C), Mψ1 = SO8 (C), spin+ ' Sym2 (StdSp2 ) ⊕ ν5 and spin− ' StdSp2 ⊗ν4 . Here u1 = +, and we conclude ρψ ' (ν7 ⊕ 1) ⊗ (Sym2 (StdSp2 ) ⊕ ν5 ) ' Sym2 (StdSp2 ) ⊗ (ν7 ⊕ 1) ⊕ ν11 ⊕ ν9 ⊕ ν7 ⊕ ν5⊕2 ⊕ ν3 . In this example, as in general, we would love to know that the above formula is valid for the rational Hodge structure Hψ• , replacing Sym2 (StdSp2 ) by Sym2 (M)(11) where M is the motivic Hodge structure associated to Δ11 . In [99] this (and generalizations) is proved at the level of -adic Galois representations. Forgetting the Hodge structure, the graded vector space Hψ• is completely described by the restriction of ρψ to SL2 (C). The Laurent polynomial T −g(g+1)/2

g(g+1)  k=0

T k dim Hψk

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can easily be computed by taking the product over 0 ≤ i ≤ r of the following Laurent polynomials (with choice of signs as in Theorem 34). Denote x = (1, diag(T , T −1 )) ∈ C× × SL2 (C). (i) For ψ0 = π0 [2d + 1] with π0 ∈ Oo (w1 , . . . , wm ), we have d    m spinψ0 ◦f (T −j + T j )2m+1 . ψ,∞ (x) = 2 j =1

(ii) For ψi = πi [2d + 1] with πi ∈ Oe (w1 , . . . , w2m ) we have d    2m−1  spin± (x) = 2 ◦ f (T −j + T j )4m . ψ,∞ ψi j =1

(iii) For ψi = πi [2d] with πi ∈ S(w1 , . . . , wm ) we have    spin± ψi ◦fψ,∞ (x) = ⎞ ⎛ d d  1 ⎝ (2 + T 2j −1 + T 1−2j )m ± (2 − T 2j −1 − T 1−2j )m ⎠ . 2 j =1

j =1

Acknowledgements Klaus Hulek presented his joint work with Sam Grushevsky [55] at the Oberwolfach workshop “Moduli spaces and Modular forms” in April 2016. During this workshop Dan Petersen pointed out that I H • (AgSat ) can also be computed using [98]. I thank Dan Petersen, the organizers of this workshop (Jan Hendrik Bruinier, Gerard van der Geer and Valery Gritsenko) and the Mathematisches Institut Oberwolfach. I also thank Eduard Looijenga for kindly answering questions related to Zucker’s conjecture.

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Syzygies of Curves Beyond Green’s Conjecture Michael Kemeny

Abstract We survey three results on syzygies of curves beyond Green’s conjecture, with a particular emphasis on drawing connections between the study of syzygies and other topics in moduli theory.

1 Introduction Arguably the central concept of modern commutative algebra is that of a minimal free resolution. Let S be either a local Noetherian ring or a positively graded, finitely generated algebra over a field. The task is then to describe the shape of the minimal free resolutions of various classes of S modules. In this survey, we restrict attention to the graded ring R = C[x0 , . . . , xm ], with grading Rd = {polynomials of degree d}. There is one particular class of graded R-modules which are the most interesting to algebraic geometers, namely those of the form  ΓX (L) := H 0 (X, L⊗n ), n≥0

where X is a projective variety, L is a very ample line bundle with m + 1 sections, and where ΓX (L) is a R ' Sym(H 0 (X, L)) module in the natural way. By the Hilbert Syzygy Theorem, any finitely generated, graded R module M has a minimal free resolution 0 ← M ← F0 ← F1 ← . . . ← Fk ← 0 of length at

M. Kemeny () Stanford University, Stanford, CA, USA © Springer Nature Switzerland AG 2018 J. A. Christophersen, K. Ranestad (eds.), Geometry of Moduli, Abel Symposia 14, https://doi.org/10.1007/978-3-319-94881-2_7

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most k ≤ m. Decomposing each term  Fi = R(−i − j )bi,j (M) j

into its graded pieces, this resolution defines invariants bi,j (M) of the module, called the Betti numbers of M. Question 1 In the case M = ΓX (L), what geometric information about X can be gleaned from the Betti numbers bi,j (X, L) := bi,j (ΓX (L))? The case which has received the most attention is when the variety is a projective curve C of genus g and L = ωC . For a projective curve, the geometric information one is most interested in is perhaps the gonality gon(C), which is the least degree d such that there exists a degree d cover C → P1 (we call such a cover a minimal pencil). In practice, one sometimes needs to replace gonality with a slightly refined invariant, called the Clifford index and defined as Cliff(C) := min {deg M −2h0 (M)+2 | M ∈ Pic(C), deg(M) ≤ g −1, h0 (M) ≥ 2}.

There is always the bound Cliff(C) ≤ gon(C) − 2, which, at least conjecturally, is an equality for most curves, [21]. Here one has the celebrated: Conjecture 1 (Green’s Conjecture, [35]) We have the following vanishing of quadratic Betti numbers: bp,2 (C, ωC ) = 0 for p < Cliff(C). This conjecture provides a sweeping generalisation of the following classical results of Noether and Petri: Theorem 1 (Noether, Petri) Assume C is not hyperelliptic. Then the canonical embedding C → Pg−1 is projectively normal. If, further, C does not admit a degree three cover of P1 , then IC/Pg−1 is generated by quadrics. The first serious approach to Green’s conjecture, due to Schreyer [42], relies on the observation that if a curve admits a minimal pencil f : C → P1 of degree d, then the canonical curve C lies on the rational normal scroll X ⊆ Pg−1 which can be geometrically described as the union of the span of the divisors f −1 (p) in Pg−1 as p ∈ P1 varies. The minimal free resolution of the scroll X can be described by an Eagon–Northcott resolution: Theorem 2 (Eagon–Northcott [18]) Let R be a ring and f : R r → R s be an R module homomorphism for r ≥ s. There is a complex ∧s f

∗ 0 ← R ←−− ∧s R r ← S1∗ ⊗ ∧s+1 R r ← . . . ← Sr−s ⊗ ∧r R r ← 0

where S = R[x1 , . . . , xs ], which is exact if and only if depth (Is (f )) = r − s + 1.

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In the theorem above, Ij (f ) denotes the ideal generated by j × j minors of f , for any 1 ≤ j ≤ s. By restriction to the canonical curve, the Eagon–Northcott resolution injects into the linear strand of the minimal resolution of the curve. This translates Green’s Conjecture into the prediction that the length of the Eagon–Northcott resolution equals the length of the linear strand of the canonical curve. Eagon–Northcott resolutions form an important class of resolutions in their own right, and the work of Buchsbaum and Eisenbud on understanding and generalising these resolutions led to several results which have been seminal to the development of modern commutative algebra. For example, the famous Criterion for Exactness came out of this: Theorem 3 (Buchsbaum-Eisenbud [10]) Let R be a ring and let f1

f2

fn

− F1 ← − F2 ← . . . ← − Fn ← 0 F• : F0 ← be a complex of free R modules. Assume 1. rank Fi = rank fi + rank fi+1 2. if I (fi ) = R then depth I (fi ) ≥ i Then F• is exact. Here I (fi ) is defined to be Irkfi (fi ). Using this criterion, Buchsbaum and Eisenbud construct resolutions generalising the Eagon–Northcott resolution in [11]. Perhaps surprisingly, some of the most effective tools for approaching Green’s Conjecture have come from geometry rather than algebra. It was observed early on by Green and Lazarsfeld that there is an intimate connection between Green’s Conjecture and the theory of K3 surfaces and moduli spaces of sheaves on such surfaces. This connection has proven to be surprisingly deep and has significantly influenced the subsequent development of K3 surface theory. As just one example, the following fundamental theorem came about as a verification of a prediction from Green’s Conjecture: Theorem 4 (Green–Lazarsfeld [37]) Let X be a K3 surface and L ∈ Pic(X) a base point free line bundle. Then Cliff(C) is constant amongst all smooth curves C ∈ |L|. In another direction, the study of syzygies of curves has proven to be remarkably important for the study of the birational geometry of the moduli space of curves, see [23, 32]. In a landmark pair of papers, Green’s Conjecture was eventually proven for a generic curve of arbitrary genus by Voisin [44, 45]. Voisin’s proof relied on a new interpretation of the problem in terms of the Hilbert scheme of points on a K3 surface:

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Theorem 5 (Voisin) Let X be a complex projective variety and L a line bundle. Then bp,1(X, L) equals the corank of the natural map [p+1] H 0 (Xcurv , det L[p+1] ) → H 0 (Ip+1 , (q ∗ det L[p+1] )|Ip+1 ) [p+1]

where Xcurv is the curvilinear locus in the Hilbert scheme of points, L[p+1] is the [p+1] tautological bundle, Ip+1 ⊆ Xcurv × X is the incidence variety and q : Ip+1 → [p+1] Xcurv is the projection. In this survey, we outline some new avenues of research going out in various directions from Green’s Conjecture, with a focus on demonstrating the connections between the study of syzygies and other aspects of moduli theory. The first of these directions is the Secant Conjecture of Green and Lazarsfeld, [36]. This conjecture gives a condition for the vanishing of quadratic syzygies of a curve embedded by a nonspecial line bundle L. The Secant Conjecture generalises the following well-known theorem of Castelnuovo–Mumford in much the same way as Green’s conjecture generalises the Theorem of Noether–Petri: Theorem 6 (Castelnuovo–Mumford) Let L be a very ample line bundle on a curve with deg(L) ≥ 2g + 1 then φL : C → Pr is projectively normal. Green and Lazarsfeld proved that one can replace the bound in Castelnuovo– Mumford’s theorem with deg(L) ≥ 2g + 1 − Cliff(C). The Secant Conjecture then extends the above results to higher syzygies. The second direction we discuss is the Prym–Green Conjecture. For a general canonical curve C → Pg−1 the generic Green’s Conjecture as proved by Voisin suffices to describe the shape of the free resolution of the homogeneous coordinate ring of C. The Prym–Green conjecture likewise predicts a similar shape for the homogeneous coordinate ring of a general paracanonical curve, that is a curve embedded by a twist of the canonical line bundle by a torsion line bundle. Lastly, we come back to Schreyer’s original approach and consider the question of whether all the syzygies in the last position of the linear strand of a canonical curve come from the syzygies of the scrolls associated to the minimal pencils of the curve. Our approach to this question is a blend of Schreyer’s original approach using the Eagon–Northcott complex with the approach of Hirschowitz–Ramanan, [39], which suggests that one should construct and study appropriate divisors on moduli spaces. In our case the moduli spaces are spaces of stable maps to Pm , which have some peculiarities in comparison with the moduli space of curves, and several natural questions are left open.

2 The Eagon–Northcott Complex It is quite rare for one to be able to construct an explicit free resolution of a module, so those few families of resolutions which we have are much prized. Perhaps the most well known and useful resolutions is given by the Koszul complex. Let R be

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a ring and {f1 , . . . , fr } a sequence of elements in R. Let f¯ : R r → R be the map sending the ith basis element ei of R r to fi for 1 ≤ i ≤ r. The Koszul complex associated to f¯ is the complex f¯

− Rr ← − ∧2 R r ← − . . . ← ∧r R r ← 0 K• (f¯) : R ← d

d

where the differential is defined by d(ei1 ∧ . . . ∧ eip ) =

p 

(−1)j +1 fij ei1 ∧ . . . ∧ eˆij ∧ . . . ∧ eip .

j =1

The Eagon–Northcott complex was the very influential discovery that one could generalise the construction of the Koszul complex and associate a complex to any R module homomorphism g¯ : R r → R s with r ≥ s. To describe how this works, we follow [19] to construct the complex as a strand of a graded Koszul complex. Consider the graded polynomial ring S := R[x1 , . . . , xs ]. We may identify S1 with R s via the standard basis. Setting F := S(−1)⊕r ' (R r ⊗R S)(−1), then g¯ defines a morphism g˜ : F → S of graded S modules. Explicitly, if e1 , . . . , er is a basis of R r , then  g (ei ⊗ 1) = g(e ¯ i ) ∈ R s ' S1 . Taking the Koszul complex of  g , we get g ) : S ← F ← ∧2 F ← . . . ← ∧r F ← 0. K• ( This is a graded complex and taking the kth graded piece of this complex yields δ

δ

K• ( g )k : Sk ← − Sk−1 ⊗R R r ← − Sk−2 ⊗ ∧2 R r ← . . . ← Sk−r ⊗ ∧r R r ← 0. Dualizing this and using the identification ∧i R r ' ∧r−i (R r )∗ we get a complex d

∗ ∗ K•∗ ( g )k : Sk−r ← − Sk−r+1 ⊗ R r ← . . . ← Sk∗ ⊗ ∧r R r ← 0. Now set k = r − s. ∗ ∗ The first s terms S−s , . . . , S−1 are all zero, so we get a complex d

∗ − S1∗ ⊗ ∧s+1 R r ← . . . ← Sr−s ⊗ ∧r R r ← 0. ∧s R r ←

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The Eagon–Northcott complex ∧s g¯

d

∗ − S1∗ ⊗ ∧s+1 R r ← . . . ← Sr−s ⊗ ∧r R r ← 0, EN(g) ¯ : R ←−− ∧s R r ←

is then obtained by extending the above complex by ∧s g¯ (it remains a complex). Let Ij (g) ¯ denote the ideal generated by the j × j minors of g. ¯ Theorem 7 ([18]) The complex EN(g) ¯ is exact if and only if depth Is (g) ¯ = r− s + 1. For example, in the special case s = 1, this gives us the well-known statement that K• (f¯) is exact if and only if {f1 , . . . , fr } forms a regular sequence. Let us now restrict attention to the graded polynomial ring R = C[x0, . . . , xm ] and set V = R1 , which is an m + 1 dimensional complex vector space. Then {x0 , . . . , xm } forms a regular sequence, and taking the Koszul complex produces the resolution 0 ← C ← R ← V ⊗C R ← ∧2 V ⊗ R ← . . . ← ∧m+1 V ⊗ R ← 0, of C ' R/(x0 , . . . , xm ). Let M be a graded R module, and consider the minimal free resolution 0 ← M ← F0 ← F1 ← . . . ← Fk ← 0. The Betti numbers of bi,j (M) are defined to be the (i + j )th graded piece of ToriR bi,j (M) := dim ToriR (M, C)i+j .  In terms of the minimal free resolution above, Fi = j R(−i−j )bi,j (M) . By tensoring the Koszul resolution of C by M and using symmetry of Tor, one immediately obtains a more concrete description of the syzygy spaces ToriR (M, C)i+j : Proposition 1 The (i, j )th syzygy space ToriR (M, C)i+j is the middle cohomology of i+1 .

V ⊗ Mj −1 →

i .

V ⊗ Mj →

i−1 .

V ⊗ Mj +1 ,

where the maps are the Koszul differentials and V = R1 .

3 Rational Normal Scrolls and Schreyer’s Approach In this section we discuss Schreyer’s approach to Green’s conjecture using rational normal scrolls. Let C be a smooth, projective curve of genus g, and let R := Sym H 0 (C, ωC ) ' C[x0, . . . , xg−1 ].

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We are interested in the Betti numbers of the graded R module ΓC (ωC ) :=



⊗n H 0 (C, ωC ).

n≥0

We define bi,j (C, ωC ) := bi,j (ΓC (ωC )). Suppose C has gonality d and let f : C → P1 be a minimal pencil, i.e. suppose φ has the minimal degree d. It was observed by Green–Lazarsfeld [35, Appendix] and Schreyer [42], that the minimal pencil imposes conditions on the possible values of the Betti numbers bp,1 (C, ωC ). In this section, we describe Schreyer’s approach. Suppose C is not hyperelliptic, so that |ωC | embeds the curve C in Pg−1 . If f : C → P1 is a minimal pencil, then for a general p ∈ P1 , the divisor f −1 (p) is a finite set of d points in Pg−1 . By geometric Riemann–Roch, the dimension of the span *f −1 (p), is given by dim*f −1 (p), = d − h0 (C, f ∗ OP1 (1)) = d − 2. Now consider the union Xf :=



*f −1 (p),.

p∈P1

Then Xf ⊆ Pg−1 is smooth, d − 1 dimensional projective variety known as a rational normal scroll which furthermore contains the curve C ⊆ Pg−1 . Rational normal scrolls have the minimal possible degree deg(X) = 1 + codimX, and as such they have been widely studied since Bertini classified varieties of minimal degree in 1907, [9]. Following [20], one may give a determinantal description of the variety Xf . Let u, v be a basis of H 0 (C, f ∗ OP1 (1)) and y1 , . . . , ys a basis of H 0 (C, ωC ⊗ f ∗ OP1 (−1)), where s = g + 1 − d. Consider the 2 × (g + 1 − d) matrix 4 3 uy1 . . . uys := A. vy1 . . . vys The canonical embedding gives an identification H 0 (C, ωC ) ' H 0 (Pg−1 , O(1)) and we may therefore consider A as a matrix of linear forms, that, is as a morphism A

R s (−1) − → R ⊕2 .

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Then Xf is defined by the locus of 2 × 2 minors of A, or, equivalently, is the locus where A does not have full rank. The homogeneous coordinate ring OXf of Xf is then the cokernel of (

2 .

∧2 A

R s )(−2) −−→

2 .

R 2 ' R.

Since Xf has codimension g − d in Pg−1 , depth I2 (A) = g − d, so the Eagon– Northcott complex gives a free resolution ∧2 A

0 ← OXf ← R ←−−

2 .

R s (−2) ← (R 2 )∗ ⊗ ∧3 R s (−3) ← Sym2 (R 2 )∗ ⊗ ∧4 R s (−4)

← . . . ← Symd−2 (R 2 )∗ ⊗ ∧s R s (−s) ← 0.

As all differentials in the above resolution are matrices with linear entries, the graded Nakayama lemma immediately implies that this Eagon–Northcott complex is minimal. In particular, the scroll Xf has Betti numbers bp,1 (Xf , O(1)) = p

% & g+1−d , p+1

with bp,q (Xf , O(1)) = 0 for q ≥ 2. The restriction OXf  ΓC (ωC ) induces injective maps Torp (OXf , C)p+1 → Torp (ΓC (ωC ), C)p+1 and hence we derive the bounds %

& g+1−d bp,1(C, ωC ) ≥ p . p+1 In particular, bg−d,1(C, ωC ) ≥ g − d. Schreyer’s Conjecture states that, under appropriate conditions, this is an equality. Conjecture 2 (Schreyer) Suppose C is a curve of gonality 3 ≤ d ≤ g+1 2 . Assume 1 C has a unique minimal pencil f : C → P and that further the Brill–Noether locus Wd1 (C) = {f ∗ O(1)} is reduced. Assume furthermore that f ∗ O(1) is the unique line bundle achieving the Clifford index. Then bg−d,1(C, ωC ) = g − d. The condition d ≤

g+1 2

is precisely that C have non-maximal gonality.

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Any curve satisfying the assumptions and conclusion of Schreyer’s Conjecture must also satisfy Green’s Conjecture. Indeed, under the assumptions Cliff(C) = d − 2 and Green’s Conjecture is the statement bd−3,2 (C, ωC ) = 0. Using Serre duality, this is equivalent to bg−d+1,1(C, ωC ) = 0. But if bg−d,1(C, ωC ) = g − d, then Torg−d (OXf , C)g−d+1 → Torg−d (ΓC (ωC ), C)g−d+1 is surjective. But then any linear relation amongst the syzygies in Torg−d (ΓC (ωC ), C)g−d+1 would also be a relation amongst syzygies in Torg−d (OXf , C)g−d+1 . As there are no such relations, we must have bg−d+1,1(C, ωC ) = 0. In [43], Schreyer verified his conjecture for general curves of gonality d with g  d. This has since been verified for general curves of gonality 3 ≤ d ≤ g+1 2 in [26]: Theorem 8 ([26]) Schreyer’s Conjecture holds for a general d-gonal curve of genus g ≥ 2d − 1, i.e. for such a curve we have bg−d,1(C, ωC ) = g − d.

4 Lattice Polarised K3 Surfaces and the Secant Conjecture Consider a projective K3 surface X ⊆ Pg of degree 2g − 2. If H ⊆ Pg is a hyperplane then the adjunction formula implies C := X ∩ H ⊆ Pg−1 ' H is a canonical curve of genus g. This simple observation has deep applications to the study of syzygies of canonical curves. To explain why this should be the case, we first introduce some new notation. Let Y be any smooth projective variety and L, M line bundles on Y with L base point free and ample. Set R := Sym H 0 (Y, L) and consider the graded R module ΓY (M, L) :=



H 0 (Y, L⊗n ⊗ M).

n∈Z

We define bi,j (Y, M, L) := bi,j (ΓY (M, L)). Proposition 2 (Hyperplane Restriction Theorem, [27, 35]) Let X be a K3 surface and let L, H be line bundles with H effective and base point free. Assume either (i) L ' OX or (ii) (H · L) > 0 and H 1 (X, qH − L) = 0 for q ≥ 0. Then for

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each smooth curve D ∈ |H | bp,q (X, −L, H ) = bp,q (D, −LD , ωD ) for all p, q. Proof Let R = Sym H 0 (X, H ). Let s ∈ H 0 (X, H ) define D. By our assumptions, we have a short exact sequence ⊗s

0 → ΓX (−L, H )(−1) −→ ΓX (L, H ) → ΓD (LD , ωD ) → 0 of R modules. Taking the (p + q)th graded piece of the long exact sequence of TorR ( , C) ⊗s

p

p

p

→ TorR (ΓX (−L, H ), C)p+q−1 −→ TorR (ΓX (−L, H ), C)p+q → TorR (ΓD (−LD , ωD ), C)p+q p−1

→ TorR (ΓX (−L, H ), C)p+q−1 → p

⊗s

p

The maps TorR (ΓX (−L, H ), C)p +q−1 −→ TorR (ΓX (−L, H ), C)p +q are zero for any p , [35, §1.6.11]. Thus p

p

p −1

TorR (ΓD (−LD , ωD ), C)p +q ' TorR (ΓX (−L, H ), C)p +q ⊕ TorR

(ΓX (−L, H ), C)p +q−1 ,

for any p . Let R := Sym H 0 (D, ωD ). We have an the exact sequence 0 → C → H 0 (X, H ) → H 0 (D, ωD ) → 0. Any splitting of this sequence induces isomorphisms p .

H 0 (X, H ) '

p−1 .

H 0 (D, ωD ) ⊕

p .

H 0 (D, ωD ).

Applying Proposition 1 one deduces p

p

p−1

TorR (ΓD (−LD , ωD ), C)p+q ' TorR (ΓD (−LD , ωD ), C)p+q ⊕TorR (ΓD (−LD , ωD ), C)p+q−1.

Thus bp,q (X, −L, H ) + bp−1,q (X, −L, H ) = bp,q (D, −LD , ωD ) + bp−1,q (D, −LD , ωD ). The claim now follows by induction on p, since b−1,q (X, −L, H ) = b−1,q (D, −LD , ωD ) = 0. The Hyperplane Restriction Theorem is very powerful due to the combination of the following two facts: 1. Thanks to Voisin’s groundbreaking work as well as the work of Aprodu–Farkas, Green’s Conjecture is now known for any curve on a K3 surface, [2, 44, 45].

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2. By the Global Torelli Theorem for K3 surfaces, given any even lattice Λ of signature (1, ρ−1) with ρ ≤ 10, there is a nonempty moduli space of K3 surfaces X with Pic(X) ' Λ, [17]. Item (2) gives a very powerful method for constructing examples of curves with prescribed properties. We will illustrate how this works with an application to the Secant Conjecture of Green–Lazarsfeld. A line bundle L on a curve is said to satisfy property (Np ) if we have the vanishings bi,j (C, L) = 0 for i ≤ p, j ≥ 2. In terms of the classical projective geometry, then φL : C → Pr is projectively normal if and only if L satisfies (N0 ), whereas the ideal IC/Pr is generated by quadrics if, in addition, it satisfies (N1 ). The line bundle L is called p-very ample if for every effective divisor D of degree p + 1 the evaluation map ev : H 0 (C, L) → H 0 (D, L|D ) is surjective. Conjecture 3 (Secant Conjecture, [36]) Let L be a globally generated line bundle of degree d on a curve C of genus g such that d ≥ 2g + p + 1 − 2h1 (C, L) − Cliff(C). Then (C, L) fails property (Np ) if and only if L is not p + 1-very ample. It is rather straightforward to see that if L is not p + 1 very ample then bp,2 (C, L) = 0, [36], [3, Theorem 4.36]. The harder part is to go in the other direction. In the case h1 (C, L) = 0, then the Secant Conjecture reduces to Green’s Conjecture, so we will focus on the case of a non-special line bundle L, i.e. one with h1 (C, L) = 0. Theorem 9 ([27]) The Secant Conjecture holds for a general curve C of genus g and a general line bundle L of degree d on C. An elementary argument shows that if C is general then the general L ∈ Picd (C) is (p + 1)-very ample if and only if d ≥ g + 2p + 3, see the introduction to [27]. Using this inequality and the fact that if L is a globally generated, nonspecial, line bundle with bp,2 (C, L) = 0 then bp−1,2(C, L(−x)) = 0 for a general x ∈ C, Theorem 9 reduces to finding a general curve C together with

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a non-special line bundle L ∈ Picd (C) with bp,2 (C, L) = 0 in the following two cases 1. g = 2i + 1, d = 2p + 2i + 4, p ≥ i − 1 2. g = 2i, d = 2p + 2i + 3, p ≥ i − 1. We construct such curves C and line bundles L using lattice polarized K3 surfaces. Consider first the odd genus case g = 2i + 1. Let Λ = Z[C] ⊕ Z[L] be a lattice with intersection pairing %

& % & (C)2 (C · L) 4i 2p + 2i + 4 = . (C · L) (L)2 2p + 2i + 4 4p + 4

Consider a general K3 surface X with PicX ' Λ. We need to prove that bp,2 (C, LC ) = 0. From the short exact sequence 0 → ΓX (−C, L) → ΓX (L) → ΓC (LC ) → 0 we get p

p

p−1

→ TorR (ΓX (L), C)p+2 → TorR (ΓC (LC ), C)p+2 → TorR (ΓX (−C, L), C)p+2 →

for R = Sym H 0 (X, L) ' Sym H 0 (C, LC ). So it suffices to prove bp,2 (X, L) = bp−1,3(X, −C, L) = 0, or, by the Hyperplane Restriction Theorem bp,2 (D, ωD ) = bp−1,3 (D, OD (−C), ωD ) = 0 where D ∈ |L| is a smooth curve. Applying the main result of [40], the genus 2p + 3 curve D has Clifford index p + 1. Hence the vanishing bp,2 (D, ωD ) = 0 follows from Green’s Conjecture. For the vanishing bp−1,3 (D, OD (−C), ωD ) = 0, we need to replace the use of Green’s Conjecture with the following important result, which follows from [34, §3] combined with [3, §2.1]: Theorem 10 ([34]) Let C be a non hyperelliptic curve of genus g and η ∈ / Cg−j −1 − Cj for some 1 ≤ j ≤ g−1 Picg−2j −1 (C) a line bundle such that η ∈ 2 . Then bj −1,3 (C, −(ωC ⊗ η), ωC ) = 0. Here Ca − Cb denotes the difference variety of line bundles which can be expressed as a difference of effective divisors of degree a and b. The theorem above implies that for a general line bundle η of degree ≤ 2, bp−1,3(D, −(ωD ⊗ η), ωD ) = 0,

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207

where the result further specifies what is meant by “general”. We apply this in our case to the line bundle L∗D (C) which has degree 2i − 2p ≤ 2. The case of even genus g = 2i is similar. In this case one uses K3 surfaces with Picard lattice % & 4i − 2 2p + 2i + 3 . 2p + 2i + 3 4p + 4

5 The Wahl Map and the Prym–Green Conjecture Consider the moduli space Rg, parametrising pairs (C, τ ) of a smooth, genus g curve together with a torsion bundle τ of order exactly . This is an irreducible moduli space which admits a compactification R g, , [12, 13]. Here are a few reasons one might be interested in Rg,: 1. Rg, can be considered as a higher genus analogue of the modular curve parametrizing elliptic curves plus -torsion line bundles and as such ought to be interesting from a number-theoretic point of view. 2. The moduli space Rg, is very closely related to the stack of -spin curves {(C, L) | L⊗ ' ωC } and the two spaces are often considered together, [12]. The space of -spin curves has important applications to Gromov–Witten theory, [49]. 3. In the case  = 2, the space Rg,2 has been much studied in relation to Abelian varieties, due to a construction of Prym which associates an Abelian variety to a point (C, τ ) ∈ Rg,2 , [41]. Let us explain (3) in more detail. There is the Prym map Pg : Rg,2 → Ag−1 defined as follows. Let (C, τ ) ∈ Rg,2 be a point and consider the associated double cover →C ν : C which has the property ν∗ OC ' OC ⊕ τ . Pushforward of divisors defines the Norm map  → Pic2g−2 (C). Nmν : Pic2g−2 (C) Then Nm−1 ν (ωC ) has two isomorphic connected components. The Abelian variety Pg (C, τ ) is the component 0 {L ∈ Nm−1 ν (ωC ) : h (C, L) = 0 mod 2}

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with principal polarization given by the Theta divisor 0 0 Θ := {L ∈ Nm−1 ν (ωC ) : h (C, L) ≥ 2, h (C, L) = 0 mod 2}.

If (C, τ ) ∈ Rg,, then the associated paracanonical curve is the embedded curve φωC ⊗τ : C → Pg−2 . In the case  = 2, Mumford noticed that there was a close relationship between the projective geometry of a general paracanonical curve and the geometry of the Prym map. Indeed, the differential of the Prym map at a point (C, τ ) ∈ Rg,2 ⊗2 ∗ dPg : H 0 (C, ωC ) → (Sym2 H 0 (C, ωC ⊗ τ ))∗

is injective if and only if the multiplication map ⊗2 Sym2 H 0 (C, ωC ⊗ τ ) → H 0 (C, ωC )

is surjective, i.e. if and only if the corresponding paracanonical curve is projectively normal. Using a degeneration argument, Beauville verified that this indeed holds for the general (C, τ ) ∈ Rg,2, provided g ≥ 6, [7]. Debarre went one step further and showed that the ideal IC/Pg−2 is generated by quadrics for (C, τ ) ∈ Rg,2 general and g ≥ 9, [16]. Using this, he was able to conclude that Pg is in fact generically injective for g ≥ 9. It is tempting to imitate Green’s conjecture and try to generalize the Beauville– Debarre results to higher syzygies. This is achieved in the following result, which answers affirmatively a conjecture of Farkas–Ludwig [30] and Chiodo–Eisenbud– Farkas–Schreyer, [14]: Theorem 11 ([29]) Let g = 2i + 5 be odd and (C, τ ) ∈ Rg, general. Then bi+1,1 (C, ωC ⊗ τ ) = bi−1,2 (C, ωC ⊗ τ ) = 0. The result above suffices to completely determine all Betti numbers bp,q (C, ωC ⊗τ ) of a general paracanonical curve of arbitrary level  and odd genus g = 2i + 5. Indeed, the Betti table, i.e. table with (q, p)th entry bp,q (C, ωC ⊗ τ ) of such a curve is 1 b1,1 0

where bp,1 =

2 b2,1 0

... ... ...

i−1 bi−1,1 0

i bi,1 bi,2

i+1 0 bi+1,2

i+2 0 bi+2,2

... ... ...

2i + 2 0 b2i+2,2

% & % & p(2i−2p+1) 2i + 4 (p+1)(2p−2i+1) 2i+4 if p ≤ i , bp,2 = if p ≥ i. 2i + 3 p+1 2i+3 p+2

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In even genus, it is known that the analogous conjecture does not always hold see [14, 15]. In order to prove Theorem 11, one might like to use the method described in the previous section and work with suitable K3 surfaces. This was successfully carried out assuming that the torsion level is large compared to the genus, [28]. When one attempts to prove the result for all levels, one immediate difficulty arises. Since the Picard group of a K3 surface is torsion free, it is difficult to construct K3 surfaces containing paracanonical curves in such a way that the torsion bundle τ comes as a pull-back of a line bundle on the surface, at least if  is arbitrary. The solution we take in [29] is to look for something which is similar to a K3 surface but for which the Picard group admits torsion. To explain how such surfaces come about naturally, we need to describe the work of Wahl on classifying curves lying on a K3 surface, [48]. Let C be a smooth, complex curve. The Wahl map 2 .

⊗3 H 0 (C, ωC ) → H 0 (C, ωC ),

is defined by the following rule. Choose analytic coordinate charts for C. A section s ∧ t is mapped under the Wahl map to the section specified by the formula (ds)t − s(dt) on these charts, see also [47]. Let C ⊆ Pg−1 be a canonical curve and let Y ⊆ Pg , denote the cone over C. Wahl discovered his map by studying the graded module of first order deformations of Y , [46]. The first interesting graded piece of this module is precisely the cokernel of the Wahl map. In particular, if the Wahl map is nonsurjective, and if furthermore a certain obstruction group vanishes, then Y may be deformed to a Gorenstein surface X ⊆ Pg−1 with ωX ' OX which is not a cone. Such a surface looks somewhat like a K3 surface, with the caveat that it may have nasty singularities. This led Wahl to conjecture: Conjecture 4 (Wahl’s Conjecture) Let C be a curve which is Brill–Noether–Petri general. Then C lies on a K3 surface if and only if the Wahl map is nonsurjective. One direction of this is relatively easy: if C lies on a K3 surface then it follows from the infinitesimal analysis above that the Wahl map is never surjective, see [46]. Also see [8] for a simple, direct proof of this fact without deformation theory.

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As stated above, Wahl’s conjecture needs a slight modification. Arbarello– Bruno–Sernesi proved the following: Theorem 12 ([5]) Let C be a Brill–Noether–Petri general curve of genus g ≥ 12. Then C lies on a projective K3 surface X ⊆ Pg , or is a limit of such curves, if and only if the Wahl map is nonsurjective. The result above is shown to be optimal in [4]. The proof of Arbarello–Bruno– Sernesi proceeds as follows. First of all, they verify that the obstruction group of [48] vanishes for a Brill–Noether–Petri general curve C. Thus if the Wahl map of C is nonsurjective, the cone Y can be deformed to some Gorenstein surface X = Y with canonical curves as hyperplane sections. Such surfaces were classified by Epema in his thesis, [22]. This boils the task down to deciding when the surfaces appearing in Epema’s list can be smoothed (the smoothing of any such surface must be a K3 surface). In particular, there is one class of surface which features prominently in [5]. These are surfaces whose desingularization is a projective bundle over an elliptic curve. Such surfaces are an excellent candidate for proving the Prym–Green conjecture for the following reasons: (1) they arise as degenerations of smooth K3 surfaces, (2) their general hyperplane sections are Brill–Noether–Petri general [33], (3) by pulling back bundles from the elliptic curve, we have an abundance of torsion line bundles to work with. More precisely, let E be an elliptic curve and set X := P(OE ⊕ η) where η ∈ Pic0 (E) is neither trivial nor torsion. Then φ : X→E admits two sections J0 and J1 corresponding to the quotients η and OE respectively. Let r ∈ E be general, set fr := φ −1 (r) and consider the linear system |gJ0 + fr |. The general curve C ∈ |gJ0 + fr | is smooth of genus g and passes through two base  denotes the blow-up at these two points, then points x ∈ J0 and y ∈ J1 . If X



KX  ' −(J0 + J1 ),

 is the resolution of where J0 , J1 are the proper transforms of J0 , J1 . The surface X g a limiting K3 surface Y ⊆ P with two elliptic singularities. Now let b ∈ E be such that τ = b − r is -torsion, write η = a − b for some a ∈ E and set L = (g − 2)J0 + fa on X. By adjunction LC ' KC + τ and we prove that the paracanonical curve (C, LC ) satisfies the Prym–Green conjecture. One of the crucial new ingredients in the proof is that there is a canonical

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degeneration of C to J0 ∪ D where D ∈ |(g − 1)J0 + fr | has genus g − 1, which allows one to make inductive arguments on the genus.

6 Divisors on Moduli Spaces and the Extremal Betti Number of a Canonical Curve It has been known for some time that some of the most interesting loci in the moduli space of curves can be constructed using syzygies. For example, consider the divisor K := {C ∈ M10 | ∃L ∈ Pic12 (C) with h0 (L) = 4 and b0,2 (C, L) = 0} in the moduli space M10 of curves of genus 10. It was shown in [32] that the closure of this locus violates the famous Slope Conjecture of Harris–Morrison. In practice, such syzygetic loci tend to give more information about the birational geometry of moduli spaces than other kinds of loci (such as Brill–Noether loci), [14, 24]. Rather than using our knowledge of syzygies of curves to describe the geometry of moduli spaces, we can also reverse the process and use cycle calculations on moduli spaces in order to obtain information about syzygies of curves. In [39], Hirschowitz–Ramanan construct determinantally a divisor K os ⊆ M2k−1 parametrising {C ∈ M2k−1 | bk−1,1(C, ωC ) = 0} and show that it coincides set-theoretically with the divisor H ur ⊆ M2k−1 parametrising {C ∈ M2k−1 | gon(C) ≤ k} studied by Harris–Mumford, [38]. Further, as divisors K os = (k − 1)H ur ∈ A1 (M2k−1 , Q). Using this, Hirschowitz and Ramanan concluded that if one assumes Green’s Conjecture holds for a general curve of odd genus g = 2k −1 (known now by Voisin [44, 45]) then Green’s Conjecture holds for any such curve of maximal gonality k + 1, and, furthermore, Schreyer’s Conjecture holds for curves of submaximal gonality k. It was discovered by Aprodu that one can use the Hirschowitz–Ramanan computation to obtain results about curves of arbitrary gonality, [1]. Let C be a curve of genus g and non-maximal gonality k ≤ " g2 # + 2. Let Wm1 (C) denote the subvariety of Picm (C) consisting of line bundles with at least two sections. We say

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C satisfies linear growth if 1 (C) ≤ n for all 0 ≤ n ≤ g − 2k + 2. dim Wk+n

Aprodu proved that the general k-gonal curve satisfies linear growth (if k is nonmaximal) and further: Theorem 13 (Aprodu) Let C satisfy linear growth. Then C satisfies Green’s Conjecture. Aprodu’s Theorem is the sharpest known result on Green’s conjecture. Further, it was a key step in verifying that Green’s Conjecture holds for curves on arbitrary K3 surfaces, [2]. Aprodu’s Theorem relies on the following trick with nodal curves (which is a variant of an argument of Voisin [44]): let C be as in the theorem, with k = gon(C), choose n = g + 3 − 2k general pairs of points (xi , yi ) ∈ C, 1 ≤ i ≤ n and let D be the nodal curve obtained from D by identifying xi and yi . Then D has genus g+n = 2(g − k + 1) − 1 and Aprodu shows that the linear growth condition implies D ∈ / H ur. By Hirschowitz–Ramanan’s calculation, this implies bg−k+1,1(D, ωD ) = 0. But, as Voisin shows, there is a natural injective map g−k+1

TorR1

g−k+1

(C, ωC )g−k+2 → TorR2

(D, ωD )g−k+2 ,

where R1 = SymH 0 (ωC ), R2 = SymH 0 (ωD ). Hence bg−k+1,1(C, ωC ) = 0 as predicted by Green’s Conjecture. We now turn back to the problem of attempting to describe the Betti table of a canonical curve of non-maximal gonality k. Recall from Sect. 3 that if C admits m minimal pencils f1 , . . . , fm then the associated scrolls Xf1 , . . . , Xfm each contribute to the syzygies of C. Recall further that the “extremal” Betti number bg−k,1(C, ωC ) was singled out by Schreyer’s conjecture to be of particular importance. We prove Theorem 14 ([25]) Let C be a smooth curve of genus g and gonality k ≤ " g+1 2 #. Assume C admits m minimal pencils, and that the pencils are infinitesimally and geometrically in general position. Assume further that C satisfies bpf-linear growth. Then bg−k,1 (C, KC ) = m(g − k). In other words, one can read off the number of minimal pencils from the last Betti number in the linear strand, under certain generality assumptions on the minimal pencils. Let us first state the meaning of the assumptions. A curve C of genus g and gonality k satisfies bpf-linear growth provided we have the dimension estimates 1 (C) ≤ m, for 0 ≤ m ≤ g − 2k + 1 dim Wk+m 1,bpf

dim Wk+m (C) < m, for 0 < m ≤ g − 2k + 1,

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1,bpf

where Wk+m (C) denotes the locus of base point free line bundles. This condition is a slight strengthening of Aprodu’s linear growth assumption. Next, the condition that the pencils are infinitesimally in general position means that the deformation theory of any subset {fσi } ⊆ {f1 , . . . , fm } of the pencils is unobstructed (modulo the PGL(2) action). More precisely, setting Fσ := (fσi ) : C → (P1 )|σ | , we require Ext2C (ΩF•σ , OC (−p − q − r)) = 0 for all subsets σ and general p, q, r ∈ C, where ΩF•σ is the cotangent complex of Fσ . Lastly, the condition that the pencils be geometrically in general position is a condition to ensure that the scrolls contribute syzygies independently into the extremal Betti number of the curve. To describe it, choose a general divisor T of degree g − 1 − k on C. Let Qf1 , . . . , Qfm be the quadrics obtained by projecting the scrolls Xf1 , . . . , Xfm away from T . We say {f1 , . . . , fm } is geometrically in general position if the set {Qf1 , . . . , Qfm } ⊆ |OPk (2)| is in general position. In practice, these three assumptions seem relatively easy to verify. For instance, they can be checked to hold for a general curve of non-maximal gonality k admitting m ≤ 2 minimal pencils. When m ≥ 3, we lack a good understanding of when the moduli space of curves with m minimal pencils is nonempty and irreducible, which makes it harder to approach this case, but computational evidence suggests, for instance, that the assumptions are verified for g = 11 and k = 6 provided 1 ≤ m ≤ 10, which excludes only the “sporadic” cases m = 12, 20 (the assumptions should be satisfied up until the first m where the moduli space of curves with m pencils becomes empty, which in this case is m = 11). We just state a few words about the proof, for more details see the forthcoming [25]. The proof works by ultimately reducing to the case g = 2k − 1 using a variant of Aprodu’s trick. The reduction is significantly more difficult than in Aprodu’s case, as we are required to work with stable maps with unstable curves on the base, and involves the notion of “twisting” for line bundles on a family of curves with central fibre a reducible curve, see [6, 31]. In the case g = 2k − 1, in our setting the role of the Koszul divisor of Hirschowitz–Ramanan is replaced with “Eagon–Northcott” divisors defined on an appropriate moduli space H (m) of stable maps C → (P1 )m with three base points. Letting H (m) → H (1), denote the projection to the first factor, these divisors push forward to give codimension m cycles EN

m

∈ Am (H (1), Q),

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where H (1) is a suitable subset of the space of degree k stable maps C → P1 from curves of genus 2k − 1 (with three base points, to account for the automorphisms of P1 ). These cycles satisfy the identity EN

m

= (k − 1)BN

m+1

where BN m+1 ∈ Am (H (1), Q) is a cycle corresponding to curves with m + 1 minimal pencils. This equation should be seen as a generalization of Hirschowitz– Ramanan’s equation [39, Prop. 3.2]: K os = (k − 1)H ur ∈ A1 (M2k−1 , Q). Acknowledgements It is a pleasure to thank the organisers of the Abel Symposium 2017 for a wonderful conference in a spectacular location. The results in this survey are joint work with my coauthor Gavril Farkas, who has taught me much of what I know about syzygies. I also thank D. Eisenbud and F.-O. Schreyer for enlightening conversations on these topics. This survey is an amalgamation of material taken from my course on syzygies in Spring 2017 as well as talks given at UCLA and Berkeley in Autumn 2017. In particular, I thank Aaron Landesman for several corrections and improvements to my course notes. I also would like to thank the referee for the careful reading.

References 1. M. Aprodu, Remarks on Szyzgies of d-gonal curves. Math. Res. Lett. 12, 387–400 (2005) 2. M. Aprodu, G. Farkas, Green’s conjecture for curves on arbitrary K3 surfaces. Comput. Math. 147(03), 839–851 (2011) 3. M. Aprodu, J. Nagel, Koszul Cohomology and Algebraic Geometry, vol. 52 (American Mathematical Society, Providence, RI, 2010) 4. E. Arbarello, A. Bruno, Rank two vector bundles on polarised Halphen surfaces and the GaussWahl map for du Val curves (2016). arXiv:1609.09256 5. E. Arbarello, A. Bruno, E. Sernesi, On hyperplane sections of K3 surfaces. Algebraic Geometry. arXiv:1507.05002 (to appear) 6. M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, M. Moeller, Compactification of strata of abelian differentials (2016). arXiv:1604.08834 7. A. Beauville, Prym varieties and the Schottky problem. Invent. Math. 41(2), 149–196 (1977) 8. A. Beauville, J.-Y. Mérindol, Sections hyperplanes des surfaces K3. Duke Math. J. 55(4), 873–878 (1987) 9. E. Bertini, Introduzione alla geometria proiettiva degli iperspazi: con appendice sulle curve algebriche e loro singolarità. Enrico Spoerri, Pisa (1907) 10. D. Buchsbaum, D. Eisenbud, What makes a complex exact? J. Algebra 25(2), 259–268 (1973) 11. D. Buchsbaum, D. Eisenbud, Generic free resolutions and a family of generically perfect ideals. Adv. Math. 18(3), 245–301 (1975) 12. A. Chiodo, Stable twisted curves and their r-spin structures. Ann. Inst. Fourier 58(5), 1635–1689 (2008) 13. A. Chiodo, G. Farkas, Singularities of the moduli space of level curves. J. Eur. Math. Soc. 19(3), 603–658 (2017) 14. A. Chiodo, D. Eisenbud, G. Farkas, F.-O. Schreyer, Syzygies of torsion bundles and the geometry of the level l modular variety over Mg . Invent. Math. 194(1), 73–118 (2013)

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GIT Versus Baily-Borel Compactification for Quartic K3 Surfaces Radu Laza and Kieran G. O’Grady

Abstract Looijenga has introduced new compactifications of locally symmetric varieties that give a complete understanding of the period map from the GIT moduli space of plane sextics to the Baily-Borel compactification of the moduli space polarized K3’s of degree 2, and also of the period map of cubic fourfolds. On the other hand, the period map of the GIT moduli space of quartic surfaces is significantly more subtle. In our paper (Laza and O’Grady, Birational geometry of the moduli space of quartic K3 surfaces, 2016. ArXiv:1607.01324) we introduced a Hassett-Keel–Looijenga program for certain locally symmetric varieties of Type IV. As a consequence, we gave a complete conjectural decomposition into a product of elementary birational modifications of the period map for the GIT moduli spaces of quartic surfaces. The purpose of this note is to provide compelling evidence in favor of our program. Specifically, we propose a matching between the arithmetic strata in the period space and suitable strata of the GIT moduli spaces of quartic surfaces. We then partially verify that the proposed matching actually holds.

1 Introduction The general context of our paper is the search for a geometrically meaningful compactification of moduli spaces of polarized K3 surfaces, and similar varieties (with Hodge structure of K3 type). While there exist well-known geometrically meaningful compactifications of moduli spaces of smooth curves and of (polarized) abelian varieties, the situation for K3’s is much murkier. The basic fact about the moduli space of degree-d polarized K3 surfaces Kd is that, as a consequence of Torelli and properness of the period map, it is isomorphic to a locally symmetric

R. Laza Stony Brook University, Stony Brook, NY, USA e-mail: [email protected] K. G. O’Grady () “Sapienza” Universitá di Roma, Roma, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2018 J. A. Christophersen, K. Ranestad (eds.), Geometry of Moduli, Abel Symposia 14, https://doi.org/10.1007/978-3-319-94881-2_8

217

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R. Laza and K. G. O’Grady

variety Fd = Γd \D, where D is a 19-dimensional Type IV Hermitian symmetric domain, and Γd is an arithmetic group. As such, Fd has many known compactifications (Baily-Borel, toroidal, etc.), but the question is whether some of these are modular (by way of comparison, we recall that the second Voronoi toroidal compactification of Ag is modular, cf. Alexeev [1]). The most natural approach to this question is to compare birational models of Kd (e.g. those given by GIT moduli spaces of plane sextic curves, quartic surfaces, complete intersections of a quadric and a cubic in P4 ) and the known compactifications of Fd via the period map. The most basic compactification of Fd is the one introduced by Baily-Borel; we denote it by Fd∗ . In ground-breaking work, Looijenga [40, 41] gave a framework for the comparison of GIT and Baily-Borel compactifications of moduli spaces of low degree K3 surfaces and similar examples (e.g. cubic fourfolds). Roughly speaking, Looijenga proved that, under suitable hypotheses, natural GIT birational models of a moduli space of polarized K3 surfaces can be obtained by arithmetic modifications from the Baily-Borel compactification. In particular, Looijenga and others have given a complete, and unexpectedly nice, picture of the period map for the GIT moduli space of plane sextics (which is birational to the moduli space of polarized K3’s of degree 2), see [10, 39, 50], and for the GIT moduli space of cubic fourfolds (which is birational to the moduli space of polarized hyper-kähler varieties of Type K3[2] with a polarization of degree 6 and divisibility 2), see [31, 32, 42]. By contrast, at first glance, Looijenga’s framework appears not to apply to the GIT moduli space of quartic surfaces (and their cousins, double EPW sextics): [51] and [47] showed that the GIT stratification of moduli spaces of quartic surfaces and EPW sextics, respectively, is much more complicated than the analogous stratification of the GIT moduli spaces of plane sextics or cubic fourfolds, and there is no decomposition of the (birational) period map to the Baily-Borel compactification into a product of elementary modifications as simple as that of the period map of degree 2 K3’s or cubic fourfolds. In our paper [34], we refined Looijenga’s work and we proved that, morally speaking, Looijenga’s framework can be successfully applied to the period map of quartic surfaces and EPW sextics. In fact, we have noted that Looijenga’s work should be viewed as an instance of the study of variation of (log canonical) models for moduli spaces (a concept that matured more recently, starting with the work of Thaddeus [56], and continued, for example, with the so-called Hassett–Keel program). This led to the introduction, in [34], of a program, which might be dubbed Hassett–Keel–Looijenga program, whose aim is to study the log-canonical models of locally symmetric varieties of Type IV equipped with a collection of Heegner divisors (in that paper we concentrated on a specific series of locally symmetric varieties and Heegner divisors, but the program makes sense in complete generality). In particular, in [34] we made very specific predictions for the decomposition into products of elementary birational modifications of the period maps for the GIT moduli spaces of quartic K3 surfaces. Our predictions are in the spirit of Looijenga [41], i.e. the elementary birational modifications are dictated by arithmetic. There are two related issues arising here: First, the various strata in the period space should correspond to geometric strata in the GIT compactification. Secondly, our work in [34] is only predictive, i.e. there is

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no guarantee that the given list of birational modifications is complete, or even that all these modifications occur. The purpose of this note is to partially address these two issues. Namely, we give what we believe to be a complete matching between the geometric and arithmetic strata, thus addressing the first issue. We view this result as strong evidence towards the completeness and accuracy of our predictions. While our previous paper [34] looks at the period map from the point of view of the target (the Baily-Borel compactification of the period space), the present paper’s vantage point is that of the GIT moduli spaces of quartic surfaces: we get what appears to be a snapshot of the predicted decomposition of the period map into a product of simple birational modifications. Let us discuss more concretely the content of this note, and its relationship to [34]. To start with, we recall that in [34] we have introduced, for each N ≥ 3, an Ndimensional locally symmetric variety F (N) associated to the D lattice U 2 ⊕DN−2 . The space F (19) is the period space of degree 4 polarized K3 surfaces, and also F (18), F (20) are period spaces for natural polarized varieties (see Sect. 2.4 for details). The main goal of that paper is to predict the behavior of the schemes F (N, β) = ProjR(F (N), λ(N) + βΔ(N)),

β ∈ [0, 1] ∩ Q,

where λ(N) is the Hodge (automorphic) divisor class on F (N), Δ(N) is a “boundary” divisor, with a clear geometric meaning for N ∈ {18, 19, 20}, and R(F (N), λ(N) + βΔ(N)) is the graded ring associated to the Q-Cartier divisor class λ(N) + βΔ(N). For all N, the scheme F (N, 0) is the Baily-Borel compactification F (N)∗ . At the other extreme, for N = 19, 18, the scheme F (N, 1) is isomorphic to a natural GIT moduli space M(N) (and we are confident that the same remains true for N = 20). From now on, we will concentrate our attention on F := F (19) (see [35] for a complete discussion of the case N = 18). The relevant GIT moduli space is that of quartic surfaces, i.e. M := |OP3 (4)|//PGL(4). The period map p : M  F ∗ is birational by Global Torelli. We expect (following Looijenga) that the inverse p−1 decomposes as the product of a Q-factorialization, a series of flips, and, at the last step, a divisorial contraction. In order to be more specific, we need to describe the boundary divisor Δ for F . First, let Hh , Hu ⊂ F be the (prime) divisors parametrizing periods of hyperelliptic degree 4 polarized K3’s, and unigonal degree 4 polarized K3’s respectively—they are both Heegner (i.e. Noether-Lefschetz) divisors. The boundary divisor is given by Δ := (Hh + Hu )/2.

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The birational transformations mentioned above are obtained by considering F (β) := F (19, β) for β ∈ [0, 1] ∩ Q. The main result of our previous paper is the prediction of the critical values of β corresponding to the flips, together with the description of (the candidates for) the centers of the flips on the F side. In fact in [34] we have defined towers of closed subsets (see (14)) Z 9 ⊂ Z 8 ⊂ Z 7 ⊂ Z 5 ⊂ Z 4 ⊂ Z 3 ⊂ Z 2 ⊂ Z 1 = supp Δ ⊂ F ,

(1)

where k denotes the codimension (Z 6 is missing, no typo). Our prediction is that the critical values of β are 1 1 1 1 1 1 1 0, , , , , , , , 1, 9 7 6 5 4 3 2

(2)

and that the center of the n-th flip (corresponding to the n-th critical value) is the closure of the strict transform of the n-th term in the relevant tower (the Qfactorialization corresponds to small β > 0, hence the corresponding 0-th critical β is 0). The last critical value of β, i.e. β = 1 corresponds to the contraction of the strict transform of the boundary divisor. On the GIT moduli space side, Shah [51] has defined a closed locus MI V ⊂ M containing the indeterminacy locus of the period maps (we predict that it coincides with the indeterminacy locus), which has a natural stratification (see Definition 5) MI V =(W8 {υ})⊃(W7 {υ})⊃(W6 {υ})⊃(W4 {υ})⊃(W3 {υ})⊃(W2 {υ})⊃(W1 {υ})⊃(W0 {υ}),

where υ is the point corresponding to the tangent developable of a twisted cubic curve, and the index denote dimension. As predicted by Looijenga, and refined by us, we expect that the center in M corresponding to the center Z k is Wk−1  {υ} (N.B. the indices represent the codimension and respectively the dimension of the corresponding loci. Since Z • and W• are related via flips, there is a shift by 1 for the indices.). The purpose of this note is to give evidence in favor of the above matching. We prove that the described matching holds for Z 1 and Z 2 (equivalently, for (W1  {υ}) and (W0  {υ})), and we provide evidence for the matching between Z 9 , Z 8 , Z 7 and (W8  {υ}), (W7  {υ}), (W6  {υ}) respectively. In Sect. 2 we give a very brief overview of the framework developed by Looijenga in order to compare the GIT and Baily-Borel compactifications of moduli spaces of polarized K3 surfaces, or similar varieties, and we will illustrate it by giving a bird’s-eye-view of the period map for degree-2 K3’s and cubic fourfolds. We then introduce the point of view developed in [34], and we describe in detail the predicted decomposition of the inverse of the period map for quartic surfaces as product of elementary birational maps (i.e. flips or contractions), see (9). We continue in Sect. 3, by revisiting the work of Shah [51] on the GIT for quartic surfaces. Usually, in a GIT analysis, by boundary one understands the locus (in the GIT quotient) parameterizing strictly semistable objects, which then can

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be stratified in terms of stabilizers of the polystable points (see Kirwan [24]). In his works on periods of quartic surfaces, Shah (see [49, 50]) noted that a more refined stratification emerges when studying the period map, resulting into four Types of quartic surfaces, labeled I–IV, with corresponding locally closed subsets of M denoted MI , . . . , MI V . A quartic is of Type I–III if it is cohomologically insignificant (or from a more modern point of view, it is semi-log-canonical), and thus the period map extends over the open subset of the moduli space parametrizing such surfaces; moreover the Type determines whether the period point belongs to the period space (Type I), or it belongs to one of the Type II or Type III boundary components of the Baily-Borel compactification. The remaining surfaces are of Type IV, in particular the indeterminacy locus of p : M  F (19)∗ is contained in MI V (we predict that it coincides with MI V ). In the analogous case of the period map from the GIT moduli space of plane sextics to the period space for polarized K3’s of degree 2, the Type IV locus consists of a single point (corresponding to the triple conic). On the other hand, for quartic surfaces the Type IV locus is of big dimension and it has a complicated structure. In our revision of Shah’s work, we shed some light on the structure of Type IV (and Type II and III) loci. While arguably everything that we do here is contained in Shah, we believe that the structure becomes transparent only after one knows the predicted arithmetic behavior. In some sense, the main point of Looijenga is to bring order to the world of GIT quotients of varieties of K3 type, by relating it to the orderly world of hyperplane arrangements. In Sect. 4, we define partitions of MI I and MI I I into locally closed subsets (our partitions are slightly finer than partitions which have already been defined by Shah in [51]), and we define the stratification of MI V discussed above. In Sects. 5 and 6 we provide evidence in favor of the predictions of [34] for p : M  F ∗ . We start (Sect. 5) by showing that the period map behaves as predicted in neighborhoods of the points υ, ω ∈ M corresponding to the tangent developable of a twisted cubic curve and a double (smooth) quadric respectively. By blowing up those points one “improves” the behavior of the period map; the exceptional divisor over υ maps regularly to the (closure of the) unigonal divisor in F ∗ , the exceptional divisor over ω maps to the (closure of the) hyperelliptic divisor Hh in F ∗ , and the image of the set of regular points for the map in Hh is precisely the complement of Z 2 . This result is essentially present in [51] (and belongs to “folk” tradition); we take care in specifying the weighted blow up that one needs to perform around υ in order to make the map regular above υ. In the language that we introduced previously, the above results match Z 1 with W0  {υ}. Next, we match Z 2 and W1  {υ}. This is the first flip in the chain of birational modifications transforming the GIT into the Baily-Borel compactification, and it is more involved than the blow-ups of υ and ω. It suffices here to mention that W1 parametrizes quartics Q1 + Q2 , where Q1 , Q2 are quadrics tangent along a smooth conic. (Warning: we do not provide full details of some of the proofs.) We note that while some similar arguments and computations occur previously in the literature (esp. in work of Shah [50, 51]), to our knowledge, the discussion here is the most complete and detailed analysis of an explicit (partial) resolution of a period map for K3 surfaces (esp. the discussion of the flip is mostly new).

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In Sect. 6, we provide evidence in favor of the matching of Z 9 , Z 8 , Z 7 and W8  {υ}, W7  {υ}, W6  {υ}. It is interesting to note that the flips of Z 9 , Z 8 , Z 7 are associated to the so-called Dolgachev singularities (aka triangle singularities or exceptional unimodular singularities) E12 , E13 , and E14 respectively. These are the simplest non-log canonical singularities, essentially analogous to cusp for curves. The geometric behavior of variation of F (β) at the corresponding critical values is 9 analogous to the behavior of the Hassett-Keel space Mg (α) around α = 11 (when stable curves with an elliptic tail are replaced by curves with cusps, see [19]). While hints of this behavior exist in the literature (see Hassett [17], Looijenga [37], and Gallardo [12]), our F (β) example is the first genuine analogue of a Hassett-Keel behavior for surfaces (the existence of this is well-known speculation among experts in the field). In the final section (Sect. 7), we discuss Looijenga’s Q-factorialization of F ∗ , , and the matching between the irreducible components of MI I that we denote F (i.e. the elements of the partition of MI I defined in Sect. 4) and the irreducible components of F I I (i.e. the Type II boundary components of F ∗ ). From our point of view, Looijenga’s Q-factorialization of F ∗ is nothing else but F () for  > 0 small (the prediction of [34] is that 0 <  < 1/9 will do). We compute of the Type II boundary components of the dimensions of the inverse images in F F ∗ . Lastly, we match the irreducible components of MI I and the Type II boundary components of F ∗ . This matching deserves a more detailed discussion elsewhere. On the GIT side, MI I has 8 components (of varying dimension), while F ∗ has 9 Type II boundary components (as computed in [48]), each of them is a modular curve. By adapting arguments of Friedman in [10], we can match each of the 8 components of MI I to one of the 9 Type II boundary components of F ∗ , and hence exactly one Type II boundary component is left out. The discrepancy of dimensions between GIT and Baily-Borel strata (for the 8 matching strata) is explained by Looijenga’s Q-factorialization of the Baily-Borel compactification (one of the main results of [41]). A mystery, at least for us, was the presence of a “missing” Type II boundary of F ∗ . This has to do with what we call the second order corrections to Looijenga’s predictions (one of the main discoveries of [34]). To conclude, we believe that while further work is needed (and small adjustments might occur), there is very strong evidence that our predictions from [34] are accurate. In any case, Looijenga’s visionary idea that the natural (or “tautological”) birational models (such as GIT) of the moduli space of polarized K3s are controlled by the arithmetic of the period space is validated in the highly non-trivial case of quartic surfaces (by contrast, in the previous known examples [2, 32, 39, 42, 43, 50] only first order phenomena were visible, and thus a bit misleading). As possible applications of our program, starting from the period domain side, one can bring structure and order to the (a priori) wild side of GIT. Conversely, starting from GIT and the work of Kirwan [24, 25], one can follow our factorization of the period map (and do “wall crossing” computations) and compute, say, the Betti numbers of F . Remark 1 In subsequent work [35], we have obtained a complete validation of the predictions of [34] for the related case of hyperelliptic quartic K3 surfaces (the case

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N = 18 in the notation of loc. cit.). The geometric matching that we obtain in [35] (e.g. strata Zhk ⊂ Fh = F (18) flipped to strata Wh,k−1 ⊂ Mh = M(18)) is parallel to the geometric matching that we discuss in this paper. The main technique of [35] is VGIT, and the methods there can be regarded as complementary to what we do in this paper. Definition 1 A K3 surface is a complex projective surface X with DuVal singularities, trivial dualizing sheaf ωX , and H 1 (OX ) = 0. We let U be the hyperbolic plane, and root lattices are always negative definite. Let Λ be a lattice, and v, w ∈ Λ. We let (v, w) be the value of the bilinear symmetric form on the couple v, w, and we let q(v) := (v, v). The divisibility of v is the positive integer div(v) such that (v, Λ) = div(v)Z. Let v be primitive (i.e. v = mw implies that m = ±1); if Λ is unimodular, then div(v) = 1, in general it might be greater than 1.

2 GIT vs. Baily-Borel for Locally Symmetric Varieties of Type IV The purpose of this section is to give a very brief account of Looijenga’s framework and our enhancement from [34] (with a focus on quartic surfaces). We start with the simplest non-trivial example that fits into Looijenga’s framework—degree-2 K3 surfaces (see [39, 50], [10, §5], and [33, §1] for a concise account). We then briefly touch on the general case, and we recall how it applies to the moduli space of cubic fourfolds. Lastly, we describe in detail our (conjectural) decomposition of the period map for quartic surfaces into a product of elementary birational modifications, see [34]. Remark 2 To the best of our knowledge, the first instance of Looijenga’s framework is in Igusa’s celebrated paper [21] on modular forms of genus 2. The paper by Igusa analyzes the (birational) period map between the compactification of the moduli space of (smooth) genus 2 curves provided by the GIT quotient of binary sextics and the Satake compactification of A2 (notice that A2 is a locally symmetric variety of Type IV). Igusa describes explicitly the blow-up of a non-reduced point in the GIT moduli space needed to resolve the period map. See [18] for a more recent version of this story.

2.1 Degree-2 K3 Surfaces Let F2 be the period space of degree-2 polarized K3 surfaces, i.e. F2 = Γ2 \D, where Γ2 and D are defined as follows. Let Λ := U 2 ⊕ E82 ⊕ A1 . Thus Λ is

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isomorphic to the primitive integral cohomology of a polarized K3 of degree 2. Then D := {[σ ] ∈ P(Λ⊗C) | q(σ ) = 0,

q(σ +σ ) > 0}+ ,

Γ2 := O + (Λ).

(3)

Here the first superscript + means that we choose one connected component (there are two, interchanged by complex conjugation), the second one means that Γ2 is the index-2 subgroup of O(Λ) which maps D to itself. Let F2 ⊂ F2∗ be the BailyBorel compactification. Let M2 := |OP2 (6)|//PGL(3) be the GIT moduli space of plane sextics. We let p : M2  F2∗ ,

p−1 : F2∗  M2

be the (birational) period map and its inverse, respectively. By Shah [50] the period map p is regular away from the point q ∈ M2 parametrizing the PGL(3)-orbit of 3C, where C ⊂ P2 is a smooth conic (a closed orbit in |OP2 (6)|ss ). Let MI2 ⊂ M2 be the open dense subset of orbits of curves with simple singularities, and let Hu ⊂ F2 be the unigonal divisor, i.e. the divisor parametrizing periods of unigonal degree2 K3’s. Thus Hu is a Heegner divisor; it is the image in F2 of a hyperplane v ⊥ ∩ D, where v ∈ Λ is such that q(v) = −2 and div(v) = 2 (any two such elements of Λ are Γ2 -equivalent). Then the period map defines an isomorphism ∼ MI2 −→ (F2 \ Hu ). Let L ∈ Pic(M2 )Q be the class induced by the hyperplane class on |OP2 (6)|, let λ be the Hodge divisor class on F2 , and Δ := Hu /2; a computation similar (but simpler) to those carried out in Sect. 4 of [34] gives that 1 p−1 L|F2 = λ + Hu = λ + Δ 2

(4)

(the 12 factor indicates that Hu is a ramification divisor of the quotient map D → F2 ). Arguing as in Sect. 4.2 of [34], one shows that p−1 is regular on all of F2 (one key point is that F2 is Q-factorial). On the other hand p−1 is not regular 2 ⊂ F ∗ × M2 on all of F2∗ . In order to describe p−1 on the boundary of F2 , let F 2 ∗ −1 be the graph of p , and let Π : F2 → F , Φ : F2 → M2 be the projections: 2

(5) Thus Π is an isomorphism over F2 (because p−1 is regular on F2 ). On the other hand, it follows from Shah’s description of semistable orbits in |OP2 (6)|,

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that the fibers of Π over two of the four 1-dimensional boundary components of F2∗ are 1-dimensional (namely those labeled by E82 ⊕ A1 and D16 ⊕ A1 ; see Remark 5.6 of [10] for the notation), and they are 0-dimensional over the remaining two boundary components. From this it follows that F2∗ is not Q-factorial, because if it were Q-factorial, the exceptional set of Π would have pure codimension 1. Moreover, it follows that Π is a Q-factorialization of F2∗ . In fact, since F2 is Q-factorial, with rational Picard group freely generated by λ and Hu , the rational class group Cl(F2∗ )Q is freely generated by λ∗ and Hu∗ (obvious notation). Since λ∗ is the class of a Q-Cartier divisor, it follows that Hu∗ is not Q-Cartier. Let 2 be the strict transform of Hu . Then H u ⊂ F u is Q-Cartier, because by (4), H there exists m  0 such that mHu is the divisor of a sectionof the line-bundle ∗ 2 with Proj( Φ ∗ L2m ⊗ Π ∗ (λ∗ )−2m . Moreover we can identify F n≥0 OF2∗ (nHu )), ∗ ∗ ∗ u is Π-ample (clearly aπ (λ ) + bΦ L is ample for any a, b ∈ Q+ , because H u is Π-ample). using (4) and the triviality of π ∗ (λ∗ ) on fibers of Π, it follows that H −1 as follows: first we construct the QThus (as in [39]) we have decomposed p  factorialization of F2∗ given by Proj( n≥0 OF2∗ (nHu∗ )), then we blow down the u . In this case the Mori chamber decomposition of strict transform of Hu∗ , i.e. H the cone {λ + βΔ | β ∈ [0, 1] ∩ Q} is very simple; there are exactly two walls, corresponding to β = 0 and β = 1.

2.2 A Quick Overview of Looijenga’s Framework Let M0 be a moduli space of (polarized) varieties which are smooth or “almost” smooth (e.g. surfaces with ADE singularities), with Hodge structure of K3 type. In particular the corresponding period space is F = Γ \D, where D is a Type IV domain or a complex ball, and Γ is an arithmetic group. An example of M0 is provided by the moduli space of degree-d polarized K3 surfaces, embedded by a suitable multiple of the polarization (one also has to specify the linearized ample line-bundle on the relevant Hilbert scheme), and F = Fd — in particular the example discussed in Sect. 2.1. We let M0 ⊂ M be a GIT compactification, and we let F ⊂ F ∗ be the Baily-Borel compactification. Let p : M  F ∗ be the period map, and assume that it is birational. Looijenga [40, 41] tackled the problem of resolving p. First, he observed that in many instances p(M0 ) = F \ supp Δ, where Δ is an effective linear combination of Heegner divisors— in the example of Sect. 2.1, one chooses Δ = Hu /2. It is reasonable to expect that M∼ = ProjR(F , λ + Δ),

(6)

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where λ is the Hodge (automorphic) Q-line bundle on F (of course here the choice of coefficients for Δ is crucial), and for a Q-line bundle L on F we let R(F , L ) be the graded ring of sections associated to L . In the example of Sect. 2.1, Eq. (6) holds by (4). On the other hand, Baily-Borel’s compactification is characterized as F ∗ = ProjR(F , λ). Thus, in order to analyze the period map, we must examine ProjR(F , λ + βΔ) for β ∈ (0, 1) ∩ Q (we assume throughout that R(F , λ + βΔ) is finitely generated). Let us first consider the two extreme cases: β close to 0 or to 1, that we denote β =  and β = (1 − ), respectively. The space := ProjR(F , λ + Δ) F constructed by Looijenga [41] as a semi-toric compactification, has the effect of making Δ Q-Cartier (notice that the period space F is Q-factorial, the problems → F ∗ is a small map—in the occur only at the Baily-Borel boundary). The map F ∗ example of Sect. 2.1 this is the map Π : F2 → F2 . At the other extreme, we expect  := ProjR(F , λ + (1 − )Δ) is a Kirwan type blow-up of the GIT quotient that M M with exceptional divisor the strict transform of Δ—in the example of Sect. 2.1 2 → M2 . this is the map Φ : F In between, we expect a series of flips, dictated by the structure of the preimage of Δ under the quotient map π : D → F . More precisely, let H := π −1 (supp Δ); then H is a union of hyperplane sections of D, and hence is stratified by closed subsets, where a stratum is determined by the number of independent sheets (“independent sheets” means that their defining equations have linearly independent differentials) of H containing the general point of the stratum. The stratification of H induces a stratification of supp Δ, where the strata of supp Δ are indexed by the “number of sheets” (in D, not in F = Γ \D). Roughly speaking, Looijenga predicts that a stratum of supp Δ corresponding to k (at least) sheets meeting (in D) is flipped to a dimension k − 1 locus on the GIT side. In the example of Sect. 2.1, the divisor H := π −1 Hu is smooth, and this is the reason why no flips appear in the resolution of p given by (5). In Sect. 2.3 we give an example in which one flip occurs. Summarizing, Looijenga predicts that in order to resolve the inverse of the period map p one has to follow the steps below: 1. Q-factorialize Δ. 2. Flip the strata of Δ defined above, starting from the lower dimensional strata, 3. Contract the strict transform of Δ. All these operations have arithmetic origin, and thus, when applicable, give a meaningful stratification of the GIT moduli space.

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2.3 Cubic Fourfolds The period space is similar to that of degree-2 polarized K3 surfaces (see (3)). Specifically, Λ is replaced by Λ := U 2 ⊕ E82 ⊕ A2 , and the arithmetic group is + (Λ ) := O(Λ  ) ∩ O + (Λ ), where O(Λ  ) is the stable orthogonal group. The O divisor Δ is Hu /2, where this time Hu is the image in F of v ⊥ ∩ D for v ∈ Λ such that q(v) = −6 and div(v) = 3. In this case at most two sheets of H := π −1 Hu meet, and correspondingly there is exactly one flip f , fitting into the diagram

Here, Φ is the blow-up of the polystable point corresponding to the secant variety of a Veronese surface. The map f is the flip of the codimension 2 locus where two sheets of H := π −1 Hu meet, and the corresponding locus in M is the curve parametrizing cubic fourfolds singular along a rational normal curve. For a detailed treatment, see [31, 32, 42].

2.4 Periods of Polarized K3’s of Degree 4 According to [34] We start by recalling notation and constructions from [34]. For N ≥ 3, let ΛN := + (ΛN ) < ΓN < O + (ΛN ) which is equal U 2 ⊕DN−2 . In [34] we defined a group O + to O (ΛN ) if N ≡ 6 (mod 8), and is of index 3 in O + (ΛN ) if N ≡ 6 (mod 8), see Proposition 1.2.3 of [34]. Next, we let DN := {[σ ] ∈ P(ΛN ⊗ C) | q(σ ) = 0, F (N) := ΓN \DN .

q(σ + σ ) > 0}+ ,

(7) (8)

(The meaning of the superscript + is as in (3).) Then F := F (19) is the period space for polarized K3’s of degree 4—we will explain the relevance of the other F (N) at the end of the present subsection. Let (X, L) be a polarized K3 surface of degree 4; we let and p(X, L) ∈ F be its period point. The hyperelliptic divisor Hh ⊂ F is the image of v ⊥ ∩D19 for v ∈ Λ19 such that q(v) = −4, and div(v) = 2 (any two such v’s are O + (Λ19 )-equivalent). Let (X, L) be a polarized K3 surface of degree 4; then p(X, L) ∈ Hh if and only if (X, L) is hyperelliptic, i.e. ϕL : X  |L|∨ is a regular map of degree 2 onto a quadric—this explains our terminology.

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The unigonal divisors Hu ⊂ F , is the image of v ⊥ ∩ D19 for v ∈ Λ such that q(v) = −4, and div(v) = 4 (any two such v’s are O + (Λ19 )-equivalent). If (X, L) is a polarized K3 surface of degree 4, then p(X, L) ∈ Hu if and only if (X, L) is unigonal, i.e. L ∼ = OX (A + 3B), where B is an elliptic curve and A is a section of the elliptic fibration |B|. We let Δ := (Hh + Hu )/2. For k ≥ 1, let Δ(k) ⊂ supp Δ be the k-th stratum of the stratification defined in Sect. 2.2, i.e. the closure of the image of the locus in H := π −1 (supp Δ) where k (at least) independent sheets of H meet. One has Δ(19) = ∅, and there is a strictly increasing ladder Δ(19)  Δ(18)  . . .  Δ(1) = (Hh  Hu ). This is in stark contrast with the cases discussed above: in fact (with analogous notation) in the case of degree 2 K3 surfaces one has Δ(k) = ∅ for k ≥ 2, and in the case of cubic fourfolds one has Δ(k) = ∅ for k ≥ 3. In fact, since for quartic surfaces there are 0-dimensional strata of Δ, strictly speaking Looijenga’s theory does not apply (see Lemma 8.1 in [41]). Our refinement in [34] takes care of this issue and, at least to first order, Looijenga’s framework still applies, as we proceed to explain. For the rest of the paper, the GIT moduli space M is that of quartic surfaces: M := |OP3 (4)|//PGL(4). Of course, we do not “see” hyperelliptic polarized K3’s of degree 4 among quartic surfaces, nor do we see unigonal polarized K3’s of degree 4—and that is where all the action takes place. Let λ be the Hodge Q-Cartier divisor class on F . The period map p : M  F ∗ (denoted p19 in [34]) is birational by Global Torelli, and it defines an isomorphism M∼ = ProjR(F , λ + Δ) by Proposition 4.1.2 of [34]. On the other hand, the Baily-Borel compactification F ∗ is identified with ProjR(F , λ). For β ∈ [0, 1] ∩ Q, we let F (β) = ProjR(F , λ + βΔ).

(9) The predictions of [34] are as follows. First, we expect that R(F , λ+βΔ) is finitely generated for all β ∈ [0, 1] ∩ Q, and that the critical values of β ∈ [0, 1] ∩ Q are

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given by 8 7 1 1 1 1 1 1 1 β ∈ 0, , , , , , , , 1 . 9 7 6 5 4 3 2

(10)

(Note: β = 1/8 is missing, no typo.) This means that for βi < β ≤ β < βi+1 , where βi , βi+1 are consecutive critical values, the birational map F (β)  F (β ) is an isomorphism. We let F (βi , βi+1 ) := F (β),

β ∈ (βi , βi+1 ) ∩ Q.

(11)

As we have already mentioned, F () is expected to be the Q-factorialization of F ∗ . On the other hand, F (1 − ) is the blow-up of M with center a scheme supported on the two points representing the tangent developable of a twisted cubic curve, and a double (smooth) quadric. For later reference we denote by υ and ω the corresponding points of M; explicitly υ := [V (4(x1x3 − x22 )(x0 x2 − x12 ) − (x1 x2 − x0 x3 )2 )], ω :=

[V ((x02

+

x12

+ x22

+ x32 )2 )].

(12) (13)

We predict that one goes from F () to F (1 − ) via a stratified flip, summarized in (9). More precisely, in [34] we have defined a tower of closed subsets Z 9 ⊂ Z 8 ⊂ Z 7 ⊂ Z 5 ⊂ Z 4 ⊂ Z 3 ⊂ Z 2 ⊂ Z 1 = Hu ∪ Hh ⊂ F ,

(14)

where k denotes the codimension (Z 6 is missing, no typo). In fact, with the notation of [34], 1. 2. 3. 4.

for k ≤ 5, Z k = Δ(k) , Z 7 = Im(f13,19 ◦ q13 : F (II2,10 ⊕ A2 ) → F ), Z 8 = Im(f12,19 ◦ m12 : F (II2,10 ⊕ A1 ) → F ), and Z 9 = Im(f11,19 ◦ l11 : F (II2,10 ) → F ) (Z 9 is one of the two components of Δ(9) ).

Let m ∈ {2, 3, . . . , 7, 9}; we predict that the birational map F (a(m),

1 1 1 )  F ( , ) m m m−1

1 (here a(m) = m+1 if m = 7, 9, a(7) = 1/9, and a(9) = 0) is a flip with center the strict transform of (the closure) of Z k , where k = m, except for m = 7, 6, in which case k = m + 1. Thus we expect that Z k is replaced by a closed Wk−1 ⊂ M of dimension k − 1. Correspondingly, we should have a stratification of the indeterminacy locus Ind(p) of the period map. Now, according to Shah, the indeterminacy locus Ind(p) is contained in the locus MI V parametrizing polystable

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quadrics of Type IV (i.e. those which do not have slc singularities, see Sect. 3.3)— and it is natural to expect that Ind(p) = MI V . The first evidence in favor of our predictions is that, as we will show, MI V has a natural stratification MI V =(W8 {υ})⊃(W7 {υ})⊃(W6 {υ})⊃(W4 {υ})⊃(W3 {υ})⊃(W2 {υ})⊃(W1 {υ})⊃(W0 {υ}),

(15) where W0 = {ω}, and each Wi is (closed) irreducible of dimension i. It is well  of a certain subscheme of known that the period map “improves” on the blow-up M M supported on {υ, ω}. More precisely, it is regular on the exceptional divisor over υ, with image the closure of the unigonal divisor Hu , it is regular on the dense open subset of the exceptional divisor over ω parametrizing double covers of P1 × P1 ramified over a curve with ADE singularities (the exceptional divisor over ω is the GIT quotient of |OP1 ×P1 (4, 4)| modulo Aut(P1 ×P1 )), mapping it to Hh \Δ(2) . This is discussed with much more detail than previously available in the literature (e.g.  with F (1−), for [51]) in Sects. 5.1 and 5.2 respectively. In Sect. 5.3 we identify M small  > 0. Section 5.4 is devoted to a proof (without full details) that the blow up  can be contracted of a suitable scheme supported on the strict transform of W1 in M to produce F (1/2). Lastly, in Sect. 6, we give evidence that Wk−1 is related to Z k as predicted, for k ∈ {7, 8, 9}. Namely, Z 9 , Z 8 , Z 7 correspond precisely to T2,3,7, T2,4,5 , T3,3,4 marked K3 surfaces respectively, while W6 , W7 , W8 correspond to the equisingular loci of quartics with E14 , E13 , E12 singularities respectively (on the GIT side). The flips replacing Wk−1 with Z k (in this range) are analogous to the semi-stable replacement that occurs for curves in the Hassett–Keel program (e.g. curves with cusps are replaced by stable curves with elliptic tails). We end the subsection by going back to F (N) for arbitrary N ≥ 3. First, there are other values of N for which F (N) is the period space for geometrically meaningful varieties of K3 type. In fact, F (18) is the period space for hyperelliptic polarized K3’s of degree 4, and F (20) is the period space for double EPW sextics [46] (modulo the duality involution), and of EPW cubes [22]. Secondly, there is a “hyperelliptic divisor” on F (N) for arbitrary N (and a “unigonal” divisor on F (N) for N ≡ 3 (mod 8)). More precisely, if N ≡ 6 (mod 8) the hyperelliptic divisor Hh (N) ⊂ F (N) is the image of v ⊥ ∩ D for v ∈ ΛN such that q(v) = −4, and div(v) = 2, if N ≡ 6 (mod 8) the definition of the hyperelliptic divisor is subtler (there is a link with the fact that [O + (ΛN ) : ΓN ] = 3). The key aspect of our analysis in [34] is that we have a tower of locally symmetric spaces f19

f20

fN

. . . → F (18) → F (19) → F (20) → . . . → F (N − 1) → F (N) → . . . (16) where F (N − 1) is embedded into F (N) as the hyperelliptic divisor Hh (N). Our paper [34] contains analogous predictions for the behavior of ProjR(F (N), λ(N)+ Δ(N)), where λ(N) is the Hodge Q-Cartier divisor class, and Δ(N) is a Q-Cartier boundary divisor class (equal to Δ for N = 19), which are compatible with the

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tower (16). Thus the period map and birational geometry of F = F (19) sits between F (18), i.e. the period space of hyperelliptic quartic surfaces, and F (20), i.e. the period space of double EPW sextics (modulo the duality involution), or equivalently that of EPW cubes.

3 GIT and Hodge-Theoretic Stratifications of M 3.1 Summary The analysis of GIT (semi)stability for quartic surfaces was carried out by Shah in [51]. In this section we will review some of his results. In particular we will go over the GIT stratification (N.B. as usual, a stratification of a topological space X is a partition of X into locally closed subsets such that the closure of a stratum is a union of strata) determined by the stabilizer groups of polystable quartics. After that, we will review Shah’s Hodge-theoretic stratification [49, 51] M = MI  MI I  MI I I  MI V

(17)

from a modern perspective (due to Steenbrink [55], Kollár, Shepherd-Barron and others [26, 29, 52]). The period map p : M  F ∗ extends regularly away from MI V , and it maps MI , MI I and MI I I to the interior F , to the union of the Type II boundary components, and to the Type III locus (a single point) respectively. A large part of this paper is concerned with the behavior of the period map for quartic surfaces along MI V . Remark 3 Shah also defined a refinement of the stratification in (17), see Theorem 2.4 of [51] (and Sect. 4 below). We will follow the notation of Theorem 2.4 of [51], with an S prefix, and with the symbol IV replacing “Surfaces with significant limit singularities”. Thus the strata will be denoted by S-I, S-II(A,i), S-II(A,ii) SIII(B,ii), S-IV(A,i), etc. We recall that the roman numerals I, II, III, IV refer to the stratum of (17) to which a stratum belongs, and the letter A (B) indicates whether the stratum is contained in the stable locus or in the properly semistable locus. We will refer to Shah’s stratification before discussing the stratification in (17); this is not an issue, because the strata are defined explicitly by Shah in terms of singularities, see Theorem 2.4 of [51].

3.2 The GIT (or Kirwan) Stratification for Quartic Surfaces Shah [51] essentially established a relation between GIT (semi)stability of a quartic surface and the nature of its singularities. In particular he proved that a quartic with ADE singularities is stable, and hence there is an open dense subset

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MI ⊂ M parametrizing isomorphism classes of polarized K3 surfaces (X, L) such that L is very ample, i.e. (X, L) is neither hyperelliptic, nor unigonal— see Theorem 1. In the present subsection the focus is on stabilizers (in SL(4)) of strictly semistable polystable quartics (a quartic is strictly semistable if it semistable and not stable, it polystable if its PGL(4)-orbit is closed in the semistable locus |OP3 (4)|ss ), and the associated stratification of M. The point of view is essentially due to Kirwan [24, 25]. Let Ms ⊂ M be the open dense subset parametrizing isomorphism classes of GIT stable quartics. Points of the GIT boundary M \ Ms parametrize isomorphism classes of semistable polystable quartics. The stabilizer of such an orbit is a positive dimensional reductive subgroup. The classification of 1-dimensional stabilizers leads to the decomposition of the GIT boundary into irreducible components. Lemma 1 Let X = V (f ) be a strictly semistable polystable quartic. Then f is stabilized by one of the following four 1-PS’s of SL(4) (up to conjugation): λ1 = (3, 1, −1, −3), λ2 = (1, 0, 0, −1), λ3 = (1, 1, −1, −1), λ4 = (3, −1, −1, −1).

(18) For i = 1, . . . , 4, let σi ⊂ M be the closed subset parametrizing polystable points stabilized by λi . Then the following hold: 1. σ1 , . . . , σ4 are the irreducible components of the GIT boundary M \ Ms . 2. The σi ’s are related to Shah’s stratification as follows: 8 component (see B, Type II, (i) in Theorem 2.4 a. σ1 is the closure of the E in [51]) of S-II(B,i). 7 component (see B, Type II, (i) in Theorem 2.4 b. σ2 is the closure of the E in [51]) of S-II(B,i). c. σ3 = S-II(B,ii). d. σ4 = S-II(B,iii). 3. dim σ1 = 2, dim σ2 = 4, dim σ3 = 2, and dim σ4 = 1. Proof This follows from Proposition 2.2 of [51] (see also Kirwan [25, §4] for a discussion focused on stabilizers). Specifically, the first 3 cases correspond to 1PS subgroups of type (n, m, −m, −n) (i.e. Case (1) in loc. cit.). Thus λ1 , λ2 , λ3 correspond to (1.1), (1.2), and (1.3) respectively in Shah’s analysis. The last case, λ4 corresponds to the cases (2.1) or (4.1) of Shah (N.B. the two cases are dual, so they result in a single case in our lemma; the previous case (1) is self-dual). It is easy to see that the other cases in Shah’s analysis can be excluded (i.e. either they lead to unstable points, or to cases that are already covered by one of λ1 , . . . , λ4 —it is possible to have a polystable orbit stabilized by another 1-PS λ, but then the stabilizer contains a higher dimensional torus, which in turn contains a conjugate of one of λ1 , . . . , λ4 ). In conclusion, the GIT boundary consists of the 4 boundary components σi as stated (they intersect, but none is included in another).

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Item (B) of Theorem 2.4 of Shah [51] describes the strictly polystable locus in the GIT compactification. It is clear (from the geometric description and proofs) that the strictly semistable locus in M is the closure of the Type II strata, i.e. M \ Ms = ∪i=1,4 σi = S-II(B,i) ∪ S-II(B,ii) ∪ S-II(B,iii). Finally, the stratum S-II(B,i) has two components corresponding to quartics with 7 singularities respectively (see [51, Thm. 2.4 (B, 8 singularities and two E two E II(i))] for precise definitions of the two cases). In order to compute the dimensions, one can write down normal forms for the quartics stabilized by the 1-PS λi . For instance, it is immediate to see that a quartic stabilized by λ4 = (3, −1, −1, −1) is of the form x0 f3 (x1 , x2 , x3 ) (i.e. the union of the cone over a cubic curve with a transversal hyperplane, or same as S-II(B,iii)). Furthermore, we can still act on this equation with the centralizer of λ4 in SL(4). In particular, with SL(3) acting on the variables (x1 , x2 , x3 ). It follows that the dimension in M of the locus of polystable points with stabilizer λ4 (i.e. σ4 ) is 1. At the other extreme, we have the case λ1 = (3, 1, −1, −3). In this case, the centralizer is the maximal torus in SL(4). There are five degree 4 monomials stabilized by λ1 , namely x0 x23 , x13 x3 , (x0 x3 )a (x1 x2 )b with a + b = 2. It follows that dim σ1 = 2. The other cases are similar. Note that σ1 , . . . , σ4 are closed subsets of M. As a general rule, subsets of M denoted by Greek letters are closed. The intersections of the components of the GIT boundary are determined by considering stabilizers that are tori of dimension larger than 1. More in general, special strata inside the σi are determined by other reductive (non-tori) stabilizers. The stratification of GIT quotients in terms of stabilizer subgroups plays an essential role in the work of Kirwan [24], and the case of hypersurfaces of low degree was analyzed in [25]. Given a quartic X, we let Stab(X) < SL(4) be the stabilizer of X, and we let Stab0 (X) < Stab(X) be the connected component of the identity. We start by noting that we have already defined two points which are GIT strata, namely υ and ω, see (12) and (13). Remark 4 Let X ⊂ P3 be the tangent developable of a twisted cubic curve, thus in suitable homogeneous coordinates the equation of X is given in the right hand side of (12). Then X is a properly semistable polystable quartic, and the corresponding point in M is denoted by υ. The group Aut0 (X) is conjugated to SL(2) embedded in SL(4) via the Sym3 representation. Remark 5 Let X ⊂ P3 be twice a smooth quadric, thus in suitable homogeneous coordinates the equation of X is given in the right hand side of (13). Then X is a properly semistable polystable quartic, and the corresponding point in M is denoted by ω. The group Aut0 (X) is conjugated to SO(4). The following result is due to Kirwan (and essentially contained also in [51]).

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Proposition 1 (Kirwan [25, §6]) Let X be a properly semistable polystable quartic. Then Aut0 (X) is one of the following (up to conjugation): 1. The trivial group {1} (i.e. X is stable). 2. One of the 1-PS’s λ1 , . . . , λ4 in (18). 3. The two-dimensional torus diag(s, t, t −1 , s −1 ) ⊂ SL(4, C). Equivalently, X = Q1 + Q2 where Q1 , Q2 are smooth quadrics meeting along 2 pairs of skew lines (special case of S-III(B,ii)). Let τ ⊂ M be the closure of the set of points representing such quartics. Then τ is a curve, and τ = σ1 ∩ σ2 ∩ σ3 (in fact τ is the intersection of any two of the σ1 , σ2 , σ3 ). 4. The maximal torus in SL(4, C). Equivalently, X is a tetrahedron (S-III(B,i)). We let ζ ∈ M be the corresponding point. Then {ζ } = σ1 ∩ σ2 ∩ σ3 ∩ σ4 . 5. SO(3, C), or equivalently X = Q1 + Q2 where Q1 , Q2 are quadrics tangent along a smooth conic (S-IV(B,ii)). This defines a curve χ ⊂ σ2 ⊂ M. The only incidence with the other strata is χ ∩ τ = {ω}. 6. SL(2, C) embedded in SL(4) via the Sym3 -representation. Then X is the tangent developable of a twisted cubic curve (special case of S-IV (B,i)), and υ is be the corresponding point in M. One has υ ∈ σ1 , and υ ∈ / σi for i ∈ {2, 3, 4}. 7. SO(4, C). Then X = 2Q, where Q is a smooth quadric (S-IV(B,iii)), and ω is the corresponding point in M. Then ω ∈ τ , and thus ω ∈ σ1 ∩ σ2 ∩ σ3 (and ω ∈ σ4 ). Proof (Elements of the Proof.) We refer to Kirwan [25, §6] for the complete proof. Here we only describe polystable quartics parametrized by τ and χ. First we consider τ . If X consists of two quadrics meeting in two pairs of skew lines, then (in suitable homogeneous coordinates) it has equation (a1 x0 x3 + b1 x1 x2 )(a2 x0 x3 + b2 x1 x2 ) = 0. Clearly this is a pencil, and we have the following two special cases: 1. the tetrahedron (case ζ ) if any of the ai or bi vanish; 2. the double quadric (case ω) if [a1 , b1 ] = [a2 , b2 ] ∈ P1 . If both ai or bi vanish simultaneously, the associated quartic is unstable and thus the two cases above are distinct. Next we consider χ. If X consists of two quadrics tangent along a conic, then (in suitable homogeneous coordinates) it has equation fa,b := (q(x0 , x1 , x2 ) + ax32 )(q(x0 , x1 , x2 ) + bx32 ) = 0,

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for [a, b] ∈ P1 (N.B. if a = b = 0, one gets the double quadric cone, which is unstable; similarly, if a = ∞ or b = ∞, one gets an unstable quadric). Note that fa,a = 0 is the equation of the double (smooth) quadric (case ω).

3.3 The Stratification by Type Shah [49], influenced by Mumford, defined the concept of “insignificant limit singularity”, and used it to study the period map for degree 2 and degree 4 K3 surfaces (see [50, 51]). One defines MI V as the subset of M parametrizing quartics with significant limit singularities. The main point is that the restriction of the period map to (M \ MI V ) is regular. Next, F ∗ = F  F II  F III ,

(19)

where F I I is the union of the Type II boundary components, and F I I I is the (unique) Type III boundary component; this is a stratification of F ∗ . Then (19) defines, by pull-back via p, strata MI , MI I and MI I I (of course MI coincides with the set that we have already defined). (Literally speaking, we will not define MI , MI I and MI I I this way.) We will give an updated view of the concept of insignificant limit singularity. Briefly, Steenbrink [55] noticed that an insignificant limit singularity is du Bois. On a different track, from the perspective of moduli, Shepherd-Barron [52] and then Kollár–Shepherd-Barron [29] noticed that the right notion of singularities is that of semi-log-canonical (slc) singularities. More recently (with [26] as the last step), it was proved that an slc singularity is du Bois. Lastly, one can check by direct inspection that Shah’s list of insignificant singularities coincides with the list of Gorenstein slc surface singularities (which are then du Bois). Of course, in the situation studied here, this is just a long-winded highbrow reproof of Shah’s results from 1979, but what is gained is a conceptual understanding of the situation. We should also point out the connection between slc singularities and GIT. On one hand, an easy observation [14, 23] shows that a quartic with slc singularities is GIT semistable. A much deeper result (due to Odaka [44, 45]), which can be viewed as some sort of converse of this, is giving a close connection between slc singularities and K-stability. Finally, K-stability should be viewed as a refined notion of asymptotic stability. We caution however that the precise connection between asymptotic stability and K-stability/slc for K3 surfaces is not known. More precisely, an example of Shepherd-Barron [52, 53] shows that for K3s of big enough degree there is no (usual) asymptotic GIT stability. The results of [58] strengthen the meaning of this failure of asymptotic stability. Nonetheless, it is still possible that a certain (weaker) asymptotic stabilization exists. We hope that our HKL program will eventually address this issue.

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3.3.1 ADE Singularities We recall that MI ⊂ M is (by definition) the subset parametrizing isomorphism classes of quartics with ADE singularities. The following identification of MI (as a quasi-projective variety) with an open subset of the projective variety F ∗ is well known: Theorem 1 The period map defines an isomorphism ∼

MI −→ F \ (Hh ∪ Hu ).

3.3.2 Insignificant Limit Singularities We recall the following important result about slc singularities. Theorem 2 (Kollár–Kovács [26], Shah [49] (for Dimension 2)) Let X0 be a projective reduced variety (not necessarily irreducible) with slc singularities. Then X has du Bois singularities. In particular, if X /B is a smoothing of X0 over a n induces an pointed smooth curve (B, 0), then the natural map H n (X0 ) → Hlim isomorphism p,q I p,q (X0 ) ∼ = Ilim

on the I p,q components of the MHS with p · q = 0. The key point (for us) of the above result is that, if the generic fiber of X /B is a (smooth) K3 surface, then the MHS of the central fiber X0 essentially determines the limit MHS associated to X ∗ /(B \ {0}). This is a result due to Shah [49] in dimension 2 and Gorenstein singularities (the case relevant for us). Steenbrink [55] connected this result to the notion of du Bois singularities. Definition 2 A reduced (not necessarily irreducible) projective surface X0 is a degeneration of K3 surfaces if it is the central fiber of a flat proper family X /B over a pointed smooth curve (B, 0) such that ωX /B ≡ 0 and the general fiber Xb is a smooth K3 surface. We say that X0 has insignificant limit singularities if X0 has semi-log-canonical singularities. Remark 6 The list of singularities baptized as insignificant limit singularities by Shah [49] coincides with the list of Gorenstein slc singularities (see [29, 52]). For a degeneration of K3 surfaces, the Gorenstein assumption is automatic. Let X0 be a degeneration of K3 surfaces with insignificant singularities. On H 2 (X0 ) we have a MHS of weight 2. Denote by hp,q the associated Hodge numbers (hp,q = dimC I p,q ). Theorem 2 gives that one, and only one, of the following 3 equalities holds: 1. h2,0 (X0 ) = 1.

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2. h1,0 (X0 ) = 1. 3. h0,0 (X0 ) = 1. 2,0 In fact this follows from the isomorphism of the theorem, and the fact that hlim + 1,0 0,0 hlim + hlim = 1 for a degeneration of K3’s.

Definition 3 Let X0 be a degeneration of K3s. 1. 2. 3. 4.

X0 X0 X0 X0

has Type I if it has insignificant limit singularities, and h2,0 (X0 ) = 1. has Type II if it has insignificant limit singularities, and h1,0 (X0 ) = 1. has Type III if it has insignificant limit singularities, and h0,0 (X0 ) = 1. has Type IV if it has significant limit singularities.

We are interested in the case of Gorenstein slc surfaces. These are classified by Kollár-Shepherd-Barron [29] and Shepherd-Barron. They are (a) ADE singularities (canonical case) r with (b) simple elliptic singularities (for hypersurfaces the relevant cases are E r = 6, 7, 8), surfaces singular along a curve, generically normal crossings (or equivalently A∞ singularities) and possibly ordinary pinch points (aka D∞ ). (c) cusp and degenerate cusp singularities. Remark 7 We note that a normal crossing degeneration without triple points is a Type II degeneration, while a normal crossing degeneration with triple points is a Type III degeneration (a triple point is a particular degenerate cusp singularity). By applying results of Shah [49] and Kulikov-Persson-Pinkham’s Theorem (see also Shepherd-Barron [52]), one obtains the following. Theorem 3 Let X0 be a degeneration of K3 surfaces with insignificant singularities. Then the following hold: i) X0 is of Type I if and only if it has ADE singularities. ii) If X0 is of Type II then X0 has a simple elliptic singularity or it is singular along a curve which is either smooth elliptic (and has no pinch points), or rational with 4 pinch points. All other singularities of X0 are rational double points (or ADE). iii) If X0 is of Type III then, with the exception of ADE and A∞ singularities, all singularities of X0 are either cusp or degenerate cusps, and at least one of these occurs. Remark 8 We recall that the Type (I, II, III) of a K3 degeneration is nothing else than the nilpotency index (1, 2, 3) for the monodromy action N(= log Ts ) on a general fiber of a degeneration X /B. Theorem 2 allows us to read the Type in terms of the central fiber X0 (as long as X0 has slc singularities). The theorem above says that furthermore the Type of the degeneration can be determined simply by the combinatorics of X0 . We point out that this fact holds much more generally—for K-trivial varieties (see esp. [30, Section 2] and [15, Theorem 3.3.3]).

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3.3.3 The Stratification and the Period Map Proposition 2 Let X0 be a quartic surface with insignificant singularities. Then X0 is GIT semistable. Proof This follows from the general fact observed by Hacking and Kim–Lee [23] (see esp. the proof of Proposition 10.2 in [14]): GIT (semi)stability (via the numerical criterion) and the log canonical threshold are computed via the same recipe, with the difference that in the case of GIT (semi)stability one allows only linear changes of coordinates (vs. analytic in the other case). Thus, the inequality needed for log canonicity implies the inequality needed for semistability. The result also follows by inspection from Shah [51] (i.e. an unstable quartic does not have slc singularities). Definition 4 We let MI , MI I , MI I I ⊂ M be the subsets of points represented by polystable quartics with insignificant limit singularities of Type I, Type II and Type III respectively (note that MI is the same subset as the previously defined MI , by Theorem 3). We let MI V ⊂ M be the subset of points represented by polystable quartics with significant limit singularities. Below is the result that was described at the beginning of the present section. Proposition 3 MI , MI I , MI I I , MI V define a stratification of M. The period map p : M → F ∗ is regular away from MI V , and p(MI ) ⊂ F ,

p(MI I ) ⊂ F I I ,

p(MI I I ) ⊂ F I I I .

(Recall that F I I is the union of the Type II boundary components of F ∗ , and F I I I is the (unique) Type III boundary component.) Before proving Proposition 3, we prove a result on the period map  p : |OP3 (4)|  F ∗ . Define subsets |OP3 (4)|I , |OP3 (4)|I I , |OP3 (4)|I I I , |OP3 (4)|I V of |OP3 (4)| by mimicking Definition 4. Lemma 2 The period map  p is regular away from |OP3 (4)|I V , and  p(|OP3 (4)|I ) ⊂ F ,

 p(|OP3 (4)|I I ) ⊂ F I I ,

 p(|OP3 (4)|I I I ) ⊂ F I I I . (20)

Proof Let X0 ∈ (|OP3 (4)| \ |OP3 (4)|I V ) be a quartic surface. Suppose that f : (B, 0) → (|OP3 (4)|, X0 ) is a map from a smooth pointed curve, and that f (B \ {0}) is contained in the locus of smooth quartics. Let pf0 : (B \ {0}) → F ∗ be the composition  p ◦ (f |B\{0} ), and let pf : B → F ∗ be the extension to B. Then pf (0) is independent of f . In fact, this follows from Theorem 2. In addition, we see that 1. if X0 ∈ |OP3 (4)|I , then pf (0) ∈ F , 2. if X0 ∈ |OP3 (4)|I I , then pf (0) ∈ F I I , 3. and if X0 ∈ |OP3 (4)|I I I , then pf (0) ∈ F I I I .

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Now suppose that X0 ∈ (|OP3 (4)| \ |OP3 (4)|I V ), and that X0 is in the indeterminacy locus of  p. Then, since |OP3 (4)| is smooth (normality would suffice), there exist smooth pointed curves (Bi , 0i ) for i = 1, 2, and maps fi : (Bi , 0i ) → (|OP3 (4)|, X0 ) such that f (Bi \ {0i }) is contained in the locus of smooth quartics, and the points pfi (0i ) (defined as above) are different, contradicting what was just stated. This proves that  p is regular away from |OP3 (4)|I V . Equation (20) follows from Items (1), (2), (3) above. Proof (of Proposition 3) First we notice that |OP3 (4)|I , |OP3 (4)|I I , |OP3 (4)|I I I , |OP3 (4)|I V define a stratification of |OP3 (4)|, because F , F I I , F I I I define a stratification of F ∗ . Let |OP3 (4)|ss ⊂ |OP3 (4)| be the open subset of GIT semistable quartics, and let π : |OP3 (4)|ss → M be the quotient map. By definition (and the remark about |OP3 (4)|I , . . . , |OP3 (4)|I V defining a stratification of |OP3 (4)|) π −1 (M \ MI V ) ⊂ (|OP3 (4)| \ |OP3 (4)|I V ). Hence p is regular away from MI V because of Lemma 2. Lastly, MI , MI I , MI I I , MI V define a stratification of M because F , F I I , F I I I define a stratification of F ∗ .

4 Shah’s Explicit Description of the Hodge Theoretic Stratification of M In the present section, we briefly review Shah’s explicit description [51, Theorem 2.4] of the strata in the Hodge theoretic stratification of M defined in the previous section. Essentially, Shah’s strata are the intersections between the Hodge theoretic strata and the GIT strata. Then, we will slightly refine Shah’s stratification of MI V , so that the refined strata match (in “reverse order”) the strata Z m in (14). In many instances the refined strata are connected components of one of Shah’s Hodge theoretic strata.

4.1 Type II Strata for M The period map extends regularly away from MI V , and maps MI I to F I I . The matching of the irreducible components of MI I and the Type II boundary components will be given in the following section (together with an explanation of the discrepancies in dimensions). For the moment being, we note that Shah identified 8 irreducible components of MI I , and that each polystable quartic X parametrized by a point of MI I has a “j -invariant”. More precisely, either X has 6 , E 7 , or E 8 ), or sing X contains an elliptic a simple elliptic singularity (of type E curve, or a rational curve with 4 pinch points. Hodge theoretically, this corresponds 2 to the fact that GrW 1 H (X0 ) = 0 (N.B.: simple Hodge theoretic considerations show that if there is more than one source of j -invariant, e.g. two simple elliptic singularities, then the j -invariants coincide).

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Proposition 4 The Type II GIT boundary MI I consists of 8 irreducible boundary components. We label these components by II(1)–II(8). Let X be a quartic surface with closed orbit corresponding to the generic point of a Type II component. Then, X has the following description: 8 ), also the generic locus in σ1 )—Sing(X) consists of two II(1) (cf. S-II(B,i, E 8 . double points of type E  II(2) (cf. S-II(B,i, E7 ), also the generic locus in σ2 )—Sing(X) consists of two 7 and some rational double points. double points of type E II(3) (cf. S-II(B,ii), also the generic locus in σ3 )—Sing(X) consists of two skew lines, each of which is an ordinary nodal curve with four simple pinch points. II(4) (cf. S-II(B,iii), also the generic locus in σ4 )—X consists of a plane and a cone 6 ). over a nonsingular cubic curve in the plane (triple point of type E  II(5) (cf. S-II(A,i))—Sing(X) consists of a double point p of type E8 and some rational double points such that no line in X passes through p. II(6) (cf. S-II(A,ii, deg 2))—Sing(X) consists of a smooth conic C and possibly some rational double points. C is an ordinary nodal curve with 4 pinch points. II(7) (cf. S-II(A,ii, deg 3))—Sing(X) consists of a twisted cubic C and possibly some rational double points. C is an ordinary nodal curve with 4 pinch points. II(8) (cf. S-II(A,ii, deg 4))—Sing(X) consists of an elliptic normal curve of degree 4 and possibly some rational double points (equivalently X is the union of two quadric surfaces that meet transversally). Furthermore, the cases II(5)–II(8) correspond to stable quartics, while the cases II(1)–II(4) to strictly semistable quartics with generic stabilizer the 1-PSs λ1 , . . . , λ4 respectively (N.B. I I (i) = σi cf. Lemma 1). Proof This is precisely Shah [51, Thm. 2.4]. The corresponding cases in Shah’s Theorem are labeled by S-II(A/B, Case). Some of Shah’s cases (e.g. Theorem 2.4 II.A.ii) have several geometric sub-cases that are labeled in an obvious way (e.g. S-II(A,ii, deg 3) corresponding to the case when Sing(X) is a twisted cubic). 8 Singularities, cf. Urabe [57]) Let us note that there Remark 9 (Quartics with E 8 singularities. The generic are two deformation classes of quartic surfaces with E 8 . The quartic S in each of these two strata has a unique singular point p, of type E minimal resolution  S → S has the following properties: i)  S is a rational surface (a consequence of iii) below); ii) the exceptional divisor D of  S → S is a smooth elliptic curve of self8 singularities); intersection −1 (this is the condition of having E  iii) (S, D) is an anticanonical pair (i.e. D ∈ | − K S |) (this is a consequence of S being a degeneration of K3 surfaces); iv)  S comes equipped with a nef and big class h s.t. h2 = 4 and h.D = 0 (i.e. S is a quartic). v) Furthermore, we can assume that the linear system associated to h contracts only D.

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It is not hard to see (e.g. [57, Prop. 1.5]) that  S is the blow-up of P2 at 10 points 2 on a smooth cubic curve C in P (and D ⊂  S is the strict transform of C). Thus, Pic  S = *, e1 , . . . , e10 ,, where  is the pull-back of OP2 (1) and e1 , . . . , e10 are the 10 exceptional divisors. The classification of the possible divisor classes h as above was done by Urabe [57, Prop. 4.3]. Up to natural symmetries, there are two distinct possibilities: (a) h = 9l − 3(e1 + · · · + e8 ) − 2e9 − e10 (b) h = 7l − 3e1 − 2(e2 + · · · + e10 ). In other words, if  S is the blow-up of P2 at 10 (general) points on a cubic curve 8 with a divisor class h as above, then S = φ|h| ( S) is a quartic in P3 with one E singularity p. The two cases are distinguished geometrically by the fact that case (a), S contains a line passing through p (with class e10 ), while in case (b) there is no such line. By construction, it is easy to see that S depends on 10 moduli in each of the cases (a) and (b)—in particular, neither of the case is a specialization of the other one. Finally, Shah’s analysis [50, Theorem 2.4] shows that the generic surface of type (a) is strictly semistable with associated minimal orbit of Type II(1) (cf. the proposition above). In case (b), the surface S is stable (Type II(5) above). 8 Singularities) Let us note that the two Remark 10 (Arithmetic of Quartics with E cases of the Remark 9 are distinguished also from an arithmetic perspective. The arguments here are standard and are contained (with full details)in Urabe [57]. First note that since  S is the blow-up of P2 at 10 points, H 2 ( S), *, , is isometric as 2 ⊥ lattice to I1,10 . Since K = −1, it follows that the lattice KH 2 ( (notation Γ in [57]) S) is an even unimodular lattice of signature (1, 9) (and thus isometric to E8 ⊕ U ). The polarization class h has norm 4 and belongs to K ⊥ ∼ = E8 ⊕ U . It is not hard to see that there are exactly (up to isometries) two choices for h that are distinguished by the isometry class of the negative definite lattice h⊥ (notation Λ in [57]) . Namely, K⊥ ⊥ hK ⊥ is either E8 ⊕ D1 (recall D1 = *−4,) or D9 . The case (a) corresponds to E8 ⊕ D1 , while the case (b) corresponds to D9 (e.g. see [57, p. 1231]).

4.2 Type III Strata for M For completeness, we list Shah’s strata contained in MI I I . By Scattone [48], there is unique Type III boundary point in F ∗ , hence the period map sends all these strata to the same point of F ∗ . Proposition 5 A polystable quartic X corresponds to a point of MI I I if and only if one of the following holds: III(1) (cf. S-III(B,iii), also case ζ )—X consists of four planes with normal crossings (the tetrahedron). This is a single point ζ ∈ M (cf. 1 (i)).

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III(2) (cf. S-III(B,ii, 4 lines), also generic locus in τ )—X consists of two, nonsingular, quadric surfaces which intersect in a reduced curve C which consists of four lines, and whose singular locus consist of 4 double points. This gives a curve τ ◦ ⊂ M (cf. 1 (ii)), where τ ◦ = τ \ {ω, ζ }. III(3) (cf. S-III(B,ii, 2 conics))—X consists of two, nonsingular, quadric surfaces which intersect in a reduced curve, C, of arithmetic genus 1. C consists of two conics such and its singular locus consists of 2 double points; the dual graph of C is homeomorphic to a circle. This case is a specialization of the case II(8) above. Stabilizer λ4 = (1, 0, 0, −1). III(4) (cf. S-III(B,i, deg 3))—Sing(X) consists of a nonsingular, rational curve of degree 3, and some rational double points. C is a strictly quasi-ordinary, nodal curve and its set of pinch points consists of two double pinch points. Each double pinch point lies on a line in X. Stabilizer λ3 = (3, 1, −1, −3). Also a specialization of the case II(7). III(5) (cf. S-III(B,i, deg 2))—Sing(X) consists of a nonsingular, rational curve of degree 2, and some rational double points. C is a strictly quasi-ordinary, nodal curve and the set of its pinch points consists of two double pinch points. Each double pinch point lies on a line in X. Stabilized by λ4 = (1, 0, 0, −1). Specialization of the case II(6). III(6) (cf. S-III(A,ii))—Sing(X) consists of a strictly quasi-ordinary nodal curve, C, and some rational double points such that no line in X passes through a double pinch point. C is a nonsingular, rational curve of degree 2. X has either two double pinch points on C or one double pinch point and two simple pinch points on C. Specialization of the case II(6). III(7) (cf. S-III(A,i))—Sing(X) consists of a double point, p, of type T2,3,r and some rational double points such that no line in X passes through p. Specialization of the case II(5). If III(1)–III(5) holds, then X is strictly semistable, if III(6) or III(7) holds, then X is stable.

4.3 Type IV Strata for M The period map is regular away from MI V , hence in order to decompose p : M  F ∗ into a composition of simple birational maps, we must study MI V . The following is a slight refinement of Shah [51, Theorem 2.4]: Proposition 6 The Type IV locus MI V decomposes in the following strata: IV(0a) (cf. S-IV(B,iii))—X consists of a non-singular quadric surface with multiplicity 2. (Case 1(iii)). The point ω ∈ M corresponding to (generic) hyperelliptic quartics. IV(0b) (cf. S-IV(B,i, deg 3))—Sing(X) consists of a nonsingular, rational curve, C, of degree 3; C is a simple cuspidal curve. The normalization of

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IV(1)

IV(2)

IV(3)

IV(4)

IV(5)

IV(6) IV(7) IV(8)

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X is nonsingular. This is the tangent developable to the twisted cubic (Case 1(iv)). The corresponding point υ ∈ M corresponds to unigonal K3s. (cf. S-IV(B,ii))—X consists of two quadric surfaces, V1 , V2 tangent along a nonsingular conic C such that V1 ∩ V2 = 2C. (Case 1(iii)). It corresponds to a curve inside M. (cf. S-IV(B,i, deg 2))—Sing(X) consists of a nonsingular, rational curve, C, of degree 2; C is a simple cuspidal curve. The normalization of X has exactly two rational double points. Stabilized by λ4 = (1, 0, 0, −1) Sing(X) consists of a nodal curve, C, and rational double points such that no line in X passes through a non-simple pinch point. C is a nonsingular, rational curve of degree 2. Every point of X on C is a double point and the set of pinch points consists of a point of type J4,∞ . Sing(X) consists of a nodal curve, C, and rational double points such that no line in X passes through a non-simple pinch point. C is a nonsingular, rational curve of degree 2. Every point of X on C is a double point and the set of pinch points consists of either a point of type J3,∞ and a simple pinch point or a point of type J4,∞ . Sing(X) consists of a double point, p, of type J3,r and some RDPs such that no line in X passes through p. This case is a specialization of Case III(7) (and then II(8)). (cf. S-IV(A,i, E14 ))—Sing(X) consists of a double point of type E14 . (cf. S-IV(A,i, E13 ))—Sing(X) consists of a double point of type E13 . (cf. S-IV(A,i, E12 ))—Sing(X) consists of a double point of type E12 .

Remark 11 There are natural inclusions IV(k) ⊂ IV(k + 1) with the exception k = 4 (N.B. IV(4) ⊂ IV(6)). For instance, we have the following adjacencies for the exceptional unimodal singularities (aka Dolgachev singularities): E14 −→ E13 −→ E12 (see [4, p. 159]). Definition 5 We define Wk = IV(k), with the following two exceptions: W0 = IV(0a), and we skip the case k = 5. Remark 12 For quartics singular along a twisted cubic, we have the inclusions I V (0b) ⊂ I I I (4) ⊂ I I (7). Remark 13 Clearly, II(1), III(1), and IV(1) form a single stratum. The degeneracy condition is that there is a line passing through p, cf. [51, Cor. 2.3 (i)]: an isolated, non rational, double point of Type 1 through which passes a line contained in X. Remark 14 Cases II(5) and its specializations III(7) and IV(6–8) were studied by Urabe [57].

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Table 1 The geometry of the variation of models F (β) Codim (i)

Critical β

(Compn’t of) corresponding Z i ⊂ F

1 1

1 1

Hh Hu

2

1 2

Δ(2)

IV(1): 2 quadrics tangent along a conic

3

1 3

Δ(3)

IV(2): double conic, cuspidal type

4

1 4

Δ(4)

IV(3): J4,∞ -locus

5

1 5

Δ(5)

IV(4): J3,0

6

1 5

Δ(6)

7

1 6

8

1 7

9

1 9

(Compn’t of) corresponding Wi−1 ⊂ M IV(0a): double quadric IV(0b): tangent developable

IV(5): J3,∞ and J3,r

Unigonal in

Δ(6)

(T3,3,4 -polarized K3)

IV(6): E14 -locus

Unigonal in

Δ(7)

(T2,4,5 -polarized K3)

IV(7): E13 -locus

Unigonal in

Δ(8)

(T2,3,7 -polarized K3)

IV(8): E12 -locus

Our predictions regarding the matching of strata in M and strata in F is summarized in the Table 1 below. Remark 15 The points IV(0a) and IV(0b)correspond to Hh and Hu respectively; this is discussed in Sections 4 and 3 of [51]. We revisit the proof in Sects. 5.2 and 5.1 respectively. The matching for β = 12 is discussed in Sect. 5.4. Finally,

in Sect.6 we

give some evidence for the matching corresponding to the case β ∈ don’t say much about the remaining cases.

1 1 1 6, 7, 9

. We

Remark 16 We recall that the locus Z 9 ⊂ F (described as the unigonal divisor inside Δ(8) ∼ = F (11)) is one of the two components of Δ(9) . With this description, the jump from 17 to 19 is less surprising: the critical β = 19 comes from having 9 independent sheets of Δ meeting along the Z 9 locus. Remark 17 While the entire framework of the paper is similar to the Hassett-Keel program for curves, the geometric analogy

withHassett-Keel is particularly striking in the case of flips occurring for β ∈ 16 , 17 , 19 . Namely, to pass from the El (l = 12, 13, 14) locus on the GIT side to the periods side, one needs to perform a KSBA semistable replacement. This is completely analogous to the stable reduction for cuspidal curves, which leads to the elliptic tail replacement (or globally to the first ps 9 )). This part is closely related to birational modification: Mg → Mg ∼ = Mg ( 11 the work of Hassett [17] (stable replacement for curves). This is expanded on in Sect. 6.

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5 The Critical Values β = 1 and β = 1/2 The point of view of this paper is somewhat dual to that of [34]. Namely, while in [34] we have given a (conjectural) decomposition of the inverse of the period map p−1 : F ∗  M based on arithmetic considerations, here we start from the other end and attempt to resolve the period map p : M  F ∗ . As is familiar to those who have studied the analogous period maps with domains the GIT moduli spaces of plane sextics [50] and cubic fourfolds [31, 32, 42], the first step towards resolving the period map p is to blow-up the most singular points, i.e. those parametrizing polystable quartics with the largest (non virtually abelian) stabilizers (see Proposition 1). There are two such points, namely υ corresponding to the tangent developable of a twisted cubic curve and ω corresponding to a smooth quadric with multiplicity 2. In Sects. 5.1 and 5.2 we discuss a suitable  −→ M with center a subscheme whose support is {υ, ω}. Theorems 4 blow-up M  and 5 give the main results regarding  p, the pull-back of the period map to M. In short, the component of the exceptional divisor mapping to υ is identified with Mu , a projective GIT compactification of the moduli space of unigonal K3 surface (see (26)), and the component of the exceptional divisor mapping to ω is identified with Mh , the GIT moduli space of (4, 4) curves on P1 × P1 . Moreover, the lifted period map  p is regular in a neighborhood of the exceptional divisor Mu , but it is definitely not regular at all points of the exceptional divisor Mh , in fact the restriction to Mh is almost as complex as p is, there is an analogous tower of closed subsets of the relevant period space, only it has 7 terms instead of 8. It is worth remarking that the image of the restriction of  p to the regular locus is the complement of Δ(2), while the image of the restriction of p to the regular locus is the complement of Hh ∪ Hu . In this sense, in going from p to  p we have improved the behavior of the period map, and moreover  p is an isomorphism in codimension  ∼ 1, while p is not. Lastly, we have an identification M = F (1 − ) (see Corollary 2). We continue in Sect. 5.4 with the analysis of the “first flip” that occurs when   F ∗ Briefly, we show that a one tries to resolve the birational map  p : M blow-up of the curve W1 (case IV(1) in Proposition 6) followed by a contraction, accounts for double covers of the quadric cone (stratum Z 2 ⊂ F ∗ in our notation). In other words, we essentially verify1 the predicted behavior of the variation of models F (β) for β ∈ (1/2 − , 1] ∩ Q.

5.1 Blow Up of the Point υ The point υ (see IV(0b) in Proposition 6) is an isolated point of the indeterminacy locus of the period map p. The behavior of p in a neighborhood of υ is analogous to that of the period map of the moduli space of plane sextics in a neighborhood of 1 Some technical issues regarding the global construction of the flip still remain, but our analysis is fairly complete.

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the orbit of 3C (see [39, 50], [33, Thm. 1.9]), where C ⊂ P2 is a smooth conic, and is treated in Section 3 of Shah [51]. Shah’s results imply that by blowing up a subscheme of M supported at υ, one resolves the indeterminacy of p in υ; the main result is stated in Sect. 5.1.5. 5.1.1 The Germ of M at υ in the Analytic Topology We will apply Luna’s étale slice Theorem in order to describe an analytic neighborhood of υ in the GIT quotient M. Let T ⊂ P3 be the twisted cubic {[λ3 , λ2 μ, λμ2 , μ3 ] | [λ, μ] ∈ P1 }, and let X be the tangent developable of T , i.e. the union of lines tangent to T . A generator of the homogeneous ideal of X is given by f := 4(x1x3 − x22 )(x0 x2 − x12 ) − (x1 x2 − x0 x3 )2 .

(21)

Thus X is a polystable quartic representing the point υ. The group PGL(2) acts on T and hence on X; it is clear that PGL(2) = Aut(X). In order to describe an étale slice for the orbit PGL(4)X at X we must decompose H 0 (P3 , OP3 (4)) into irreducible SL2 -submodules. For d ∈ N, let V (d) be the irreducible SL2 -representation with highest weight d i.e. Symd V (1) where V (1) is the standard 2-dimensional SL2 representation. A straightforward computation gives the decomposition H 0 (OP3 (4)) ∼ = V (0) ⊕ V (4) ⊕ V (6) ⊕ V (8) ⊕ V (12).

(22)

The trivial summand V (0) is spanned by f , and the projective tangent space at V (f ) to the orbit PGL(4)V (f ) ⊂ |OP3 (4)| is equal to P(V (0) ⊕ V (4) ⊕ V (6)). We have a natural map V (8) ⊕ V (12)//SL(2) −→ M, [g] → [V (f + g)]

(23)

mapping [0] to υ. By Luna’s étale slice Theorem, the map is étale at [0]. In particular we have an isomorphism of analytic germs ∼

(V (8) ⊕ V (12)//SL(2), [0]) −→ (M, υ).

(24)

5.1.2 Moduli and Periods of Unigonal K3 Surfaces Let Ω := S• (V (8)∨ ⊕ V (12)∨ ),

(25)

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and define a grading of Ω as follows: non zero elements of V (8)∨ have degree 2, non zero elements of V (12)∨ have degree 3. Then SL(2) acts on Proj Ω, and OProjΩ (1) is naturally linearized; let Mu := Proj Ω//SL(2).

(26)

Shah (see Theorem 4.3 in [50]) proved that Mu is a compactification of the moduli space for unigonal K3 surfaces, i.e. there is an open dense subset MIu ⊂ Mu which is the moduli space for such K3’s. Moreover, the period map is regular pu

Mu −→ FII2,18 (O + (II2,18 ))∗ ,

(27)



and it defines an isomorphism MIu −→ FII2,18 (O + (II2,18 )). We recall that we have a natural regular map FII2,18 (O + (II2,18 ))∗ −→ F ∗ ,

(28)

whose restriction to FII2,18 (O + (II2,18)) is an isomorphism onto the unigonal divisor Hu , see Subsection 1.5 of [34].

5.1.3 Weighted Blow-Up We recall the construction of the weighted blow up in the case where the base is smooth. We refer to [3, 27] for details. Let (x1 , . . . , xn ) be the standard coordinates on An . Let (a1 , . . . , an ) ∈ Nn+ , and let σ be the weight given by σ (xi ) = ai . The weighted blow-up Blσ (An ) with weight σ is a toric variety defined as follows. Let {e1 , . . . , en } be the standard basis of Rn , and C ⊂ Rn be the convex cone spanned by e1 , . . . , en , i.e. the cone of (x1 , . . . , xn ) with non-negative entries. Let v := (a1 , . . . , an ) ∈ Rn , and for i ∈ {1, . . . , n} let Ci ⊂ C be the convex cone spanned by e1 , . . . , ei−1 , ei+1 , . . . , en and v. The Ci ’s generate a fan in Rn ; Blσ (An ) is the associated toric variety. Since the Ci ’s define a cone decomposition of C, we have a natural regular map πσ : Blσ (An ) → An , which is an isomorphism over An \ {0}. Let Eσ ⊂ Blσ (An ) be the exceptional set of πσ ; then Eσ is isomorphic to the weighted projective space P(a1 , . . . , an ). We denote by [x1 , . . . , xn ] (with (x1 , . . . , xn ) = (0, . . . , 0)) a (closed) point of P(a1 , . . . , an ); thus [x1 , . . . , xn ] = [y1 , . . . , yn ] if and only if there exists t ∈ C∗ such that xi = t ai yi for i ∈ {1, . . . , n}. The composition πσ

Blσ (An ) −→ An  P(a1 , . . . , an ) p → πσ (p) = (x1 , . . . , xn ) → [x1 , . . . , xn ]

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is regular; this follows from the formulae for πσ that follow Definition 2.1 in [3]. Thus we have a regular map Blσ (An ) −→ An × P(a1 , . . . , an ).

(29)

Let μσ : Eσ → P(a1 , . . . , an ) be the restriction to Eσ of the map in (29), followed by projection to the second factor. Then μσ is an isomorphism; we will identify Eσ with P(a1, . . . , an ) via μσ . The formulae for πσ that follow Definition 2.1 in [3] give the following result. Proposition 7 Keep notation as above, and let Δ ⊂ C be a disc centered at 0. Let α : Δ → Blσ (An ) be a holomorphic map such that α −1 (Eσ ) = {0}. There exists k > 0 such that πσ ◦ α(t) = (t ka1 · ϕ1 , . . . , t kan · ϕn )

(30)

where ϕi : Δ → C is a holomorphic function, and moreover α(0) = [ϕ1 (0), . . . , ϕn (0)].

(31)

(In particular (ϕ1 (0), . . . , ϕn (0)) = (0, . . . , 0).) Corollary 1 Let Z be a projective variety, and p : Blσ (An )  Z be a rational map, regular away from Eσ . Suppose that the following holds. Given a disc Δ ⊂ C centered at 0, and a holomorphic map α : Δ → Blσ (An ) such that α −1 (Eσ ) = {0}, the extension at 0 of the map p◦α|(Δ\{0}) depends only on α(0) = [ϕ1 (0), . . . , ϕn (0)] (notation as in (31)). Then p is regular everywhere. Proof Follows from Proposition 7 and normality of Blσ (An ).

5.1.4 Blow-Up of the étale Slice and the Period Map It will be convenient to denote by Z the affine scheme V (8) ⊕ V (12), i.e. Z := Spec S• (V (8)∨ ⊕ V (12)∨ ). Let (x1 , . . . , x22 ) be coordinates on V (8) ⊕ V (12) such that V (8) has equations 0 = x10 = . . . = x22 , and V (12) has equations 0 = x1 = . . . = x9 . Let σ be the weight defined by σ (xi ) :=

2 4 6

if i ∈ {1, . . . , 9}, if i ∈ {10, . . . , 22}.

(32)

 := Blσ (Z) be the corresponding weighted blow up, and let E be the Let Z  → Z; thus E is the weighted projective space P(49, 613 ). The exceptional set of Z  (and on the ample line-bundle OZ(−E)). action of SL2 on Z lifts to an action on Z

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 /SL2 . The map Z  → Z induces a map Thus there is an associated GIT quotient Z/  /SL2 −→ Z//SL2 .  μ : Z/

(33)

Moreover the set-theoretic inverse image  μ−1 ([0])red is isomorphic to Proj Ω//SL2 = Mu . Since the natural map Z//SL2 → M is dominant, it makes sense to compose it with the (rational) period map p : M  F (19)∗ . Composing with  μ, we get a rational map  /SL2  F (19)∗ .  p : Z/

(34)

Theorem 4 With notation as above, the map  p is regular in a neighborhood of μ−1 ([0])red is equal to the period map pu  μ−1 ([0])red = Mu , and its restriction to  in (28). Proof This follows from the results of Shah in [51]. More precisely, let (F, G) ∈ V (8) ⊕ V (12) be non-zero and such that [(F, G)] ∈ Proj Ω is SL2 -semistable. Let Δ ⊂ C be a disc centered at 0, and ϕ

Δ −→ V (8) ⊕ V (12) t → (t 4m F (t), t 6m G(t))

(35)

where m > 0, F (t), G(t) are holomorphic, and F (0) = F , G(0) = G. (This is the family on the second-to-last displayed equation of p. 293, with the difference that our (0, 0) ∈ Z corresponds to Shah’s F0 .) We assume also that for t = 0, the point [ϕ(t)] is not in the indeterminacy locus of the period map Z//SL2  F (19)∗ . Let pϕ : Δ → F (19)∗ be the holomorphic extension of the composition (Δ \ {0}) → Z//SL2  F (19)∗ . Then by Theorem 3.17 of [51], the value pϕ (0) is equal to the period point pu ([(F, G)]). By Corollary 1 it follows that p is regular in a neighborhood of  μ−1 ([0])red = Mu , and that the restriction of the period map −1 to  μ ([0])red is equal to the period map pu in (28). 5.1.5 Blow-Up of M at υ A weighted blow up Blσ (An ) → An is equal to the blow up of a suitable scheme supported at 0, see Remark 2.5 of [3]. It follows that also the map in (33) is the blow up of an ideal J supported on [0]. Since the map in (24) is an isomorphism of analytic germs, the ideal sheaf J defines an ideal sheaf in OM , cosupported at υ, that we will denote by I . Let Mυ := BlI M, and let Eυ ⊂ Mυ be the (reduced) exceptional divisor of BlI M → M. Thus Eυ ∼ = Mu , and Eυ is QCartier. Let φυ : Mυ → M be the natural map. By Theorem 4, the period map Mυ  F (19)∗ is regular in a neighborhood of Eυ . Moreover, letting L be the ample Q-line bundle on M descended from the ample generator of Picard group of

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∼ |O 3 (4)|, the line-bundle φ ∗ L (−E(υ)) is ample for  the parameter space P34 = P υ positive and sufficiently small.

5.2 Blow Up of the Point ω 5.2.1 The GIT Moduli Space for K3’s Which Are Double Covers of P1 × P1 The GIT moduli space that we will consider is Mh := |OP1 (4)  OP1 (4)|//Aut(P1 × P1 ).

(36)

Given D ∈ |OP1 (4)  OP1 (4)|, we let π : XD → P1 × P1 be the double cover ramified over D, and LD := π ∗ OP1 (1)  OP1 (1). If D has ADE singularities, then (XD , LD ) is a hyperelliptic quartic K3. We recall that if (X, L) is a hyperelliptic quartic K3 surface, the map ϕL associated to the complete linear system |L| ∼ = P3 is regular, and it is the double cover of an irreducible quadric Q, branched over a divisor B ∈ |OQ (4)| with ADE singularities. Vice versa, the double cover of an irreducible quadric surface, Q ⊂ P3 , branched over a divisor B ∈ |OQ (4)| with ADE singularities is a hyperelliptic quartic K3 surface. The period space for Mh is Fh ; we let ph : Mh  Fh∗

(37)

be the extension of the period map to the Baily-Borel compactification. Theorem 5 1. A divisor in |OP1 (4)  OP1 (4)| with ADE singularities is Aut(P1 × P1 ) stable, hence there exists an open dense subset MIh ⊂ Mh parametrizing isomorphism classes of hyperelliptic quartic K3 surfaces such that ϕL (X) is a smooth quadric. 2. The period map ph defines an isomorphism between MIh and the complement of the “hyperelliptic” divisor Hh (Fh ) in Fh (the divisor Hh (18) ⊂ F (18) in the notation of [34]). Proof Item (1) is a result of Shah, in fact it is contained in Theorem 4.8 of [51]. Item (2) follows from the discussion above. In fact let y ∈ Fh . Then there exists a hyperelliptic quartic K3 surface (X, L) (unique up to isomorphism) whose period point is y, and the quadric Q := ϕL (X) is smooth if and only if y ∈ / Hh (Fh ). 5.2.2 The Germ of M at ω in the Analytic Topology Let q ∈ H 0 (P3 , OP3 (2)) be a non degenerate quadratic form, and let Q ⊂ P3 be the smooth quadric with equation q = 0. Let O(q) be the associated orthogonal group;

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then PO(q) = AutQ is the stabilizer of [q 2] ∈ |OP3 (4)|. We have a decomposition of H 0 (P3 , OP3 (4)) into O(q)-modules H 0 (P3 , OP3 (4)) = q · H 0 (P3 , OP3 (2)) ⊕ H 0 (Q, OQ (2)). Note that the first submodule is reducible (it contains a trivial summand, spanned by q 2 ), while the second one is irreducible. We identify Q with P1 × P1 , PO(q) with Aut(P1 × P1 ), and H 0 (Q, OQ (2)) with H 0 (P1 × P1 , OP1 (4)  OP1 (4)). The projectivization of q · H 0 (P3 , OP3 (2)) is equal to the projective (embedded) tangent space at [q 2 ] of the orbit PGL(4)[q 2]. Thus, by Luna’s étale slice Theorem, we have natural étale map H 0 (Q, OQ (2))//O(q) −→ M, mapping [0] to ω. In particular we have an isomorphism of analytic germs ∼

(H 0 (Q, OQ (2))//O(q), [0]) −→ (M, ω).

(38)

5.2.3 Partial Extension of the Period Map on the Blow Up of ω The map φυ : Mυ → M is an isomorphism over M \ {υ}; abusing notation, we denote by the same symbol ω the unique point in Mυ lying over ω ∈ M. Let  −→ Mυ be the blow-up of the reduced point ω, and let Eω ⊂ M  be the φω : M exceptional divisor. We let φ := φυ ◦ φω , and  p = p ◦ φ. Thus we have

Proposition 8 Keeping notation as above, Eω is naturally identified with the hyperelliptic GIT moduli space Mh , and the restriction of  p to Eω is equal to the period map ph of (37). Proof Let ψω : Mω → M be the blow-up of the reduced point ω, and let pω : Mω → F ∗ be the composition p ◦ ψω . Since ω and υ are disjoint subschemes of M, the exceptional divisor of ψω is identified with Eω , and it suffices to prove  and  that the statement of the proposition holds with M p replaced by Mω and pω respectively. Let D ⊂ |OP3 (4)| be the closed subset of double quadrics, i.e. the closure of the orbit PGL(4)(2Q), where Q ⊂ P3 is a smooth quadric. Let π : P → |OP3 (4)| be the blow up of (the reduced) D, and let ED ⊂ P be the exceptional divisor of π. Then PGL(4) acts on P (because D is PGL(4)-invariant), and the

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action lifts to an action on the line bundle OP (ED ). Let L be the hyperplane line bundle on |OP3 (4)|, and let t ∈ Q+ be such that f ∗ L (−tED ) is an ample Q-line bundle on P . Then PGL(4) acts on the ring of global sections R(P , π ∗ L (−tED )), and hence we may consider the GIT moduli space   M(t) := Proj R(P , π ∗ L (−tED ))PGL(4) . By Kirwan [24], there exists t0 > 0 such that the blow down map π : P → |OP3 (4)| and (t) : M(t) induces a regular map ψ → M for all 0 < t < t0 , and moreover M(t) (t) are identified with Mω and ψω respectively. But now the identification of Eω ψ with the hyperelliptic GIT moduli space Mh follows at once from the isomorphism of germs in (38). The assertion on the period map follows from the description of the germ (M, ω) and a standard semistable replacement argument.

 5.3 Identification of F (1 − ) and M Let L be the Q line bundle on M induced by the hyperplane line bundle on |OP3 (4)|, and let L := φ ∗ L . Let E := Eυ + Eω . Then ( p−1 )∗ (E)|F = Hh + Hu = 2Δ.

(39)

In fact we have the set-theoretic equalities  p(Eυ ) ∩ F = Hu , and  p(Eω \ Ind( p)) ∩ (2) F = Hh \ Hh , thus in order to finish the proof of (39) one only needs to compute multiplicities; they are equal to 1 because  p−1 has degree 1. By (39) and Equation (4.1.2) of [34], we get ( p−1 )∗ (L(−E))|F ∼ = OF (λ + (1 − 2)Δ). Thus  p−1 induces a homomorphism R(M, L(−E)) −→ R(F , λ + (1 − 2)Δ).

(40)

Proposition 9 The homomorphism in (40) is an isomorphism of rings. Proof This is because  p−1 is an isomorphism between F \ Hh(2), which has  which again has complement of codimension 2 in F , and an open subset of M  complement of dimension 2 in M. Corollary 2 The restriction of  p−1 to F defines an isomorphism  Proj(F , λ + (1 − )Δ) ∼ =M for small enough  > 0.

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 and hence Proof If  > 0 is small enough, then L(−E) is ample on M,  ProjR(M, L(−E)) ∼ = M. Thus the corollary follows from Proposition 9.

5.4 The First Flip of the GIT Quotient (β = 1/2) We recall that the curve W1 ⊂ M contains the point ω and does not contain υ. We let  be the strict transform of W1 . We will perform a surgery of M  along W 1 ⊂ M 1 in W order to obtain our candidate for F (1/3, 1/2), notation as in (11). More precisely, → M,  which is an isomorphism we will start by constructing a birational map M 1 , and over W 1 is a weighted blow along normal slices to W 1 . Let E1 away from W → M;  then E1 ∼ 1 × Mc , where Mc is a GIT be the exceptional divisor of M =W compactification of the moduli space of degree-4 polarized K3 surfaces which are double covers of a quadric cone with branch divisor not containing the vertex of the  F be the period map and M reg ⊂ M be the subset of regular cone. Let p: M reg ∩{p}×Mc (here 1 , then the intersection M points of p; we will show that, if p ∈ W {p} × Mc ⊂ E1 ) coincides with the set of regular points of the period map Mc  F , and that the restriction of p is equal to the period map Mc  F . It follows that p is constant on the slices {p} × Mc ⊂ E1 , and the image of the restriction of p to the set of regular points of E1 is the complement of Δ(3) = Im(f16,19 ) in can be the codimension-2 locus Δ(2) = Im(f17,19) (notation as in [34]). Now, M contracted along E1 → Mc , let M1/2 be the contraction; the results mentioned above strongly suggest that M1/2 is isomorphic to F (1/3, 1/2).

5.4.1 The Action on Quartics of the Automorphism Group of polystable Surfaces in W1 Let q := x02 + x12 + x22 , and let fa,b := (q + ax32)(q + bx32 ),

(41)

where (a, b) = (0, 0). Then V (fa,b ) is a polystable quartic, and its equivalence class belongs to W1 . Conversely, if V (f ) is a polystable quartic whose equivalence class belongs to W1 , then up to projectivities and rescaling, f = fa,b for some (a, b) = (0, 0). The points in M representing V (fa,b ) and V (fc,d ) are equal if and only if [a, b] = [c, d], or [a, b] = [d, c]. Lastly, V (fa,b ) represents ω if and only if a = b.

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Suppose that a = b. Then every element of AutV (fa,b ) fixes V (x3 ) and the point [0, 0, 0, 1]. It follows that AutV (fa,b ) is equal to the image of the natural map O(q) → PGL(4). In particular SO(q) is an index 2 subgroup of AutV (fa,b ), and hence the double cover of SO(q), i.e. SL2 , acts on V (fa,b ). The decomposition into irreducible representations of the action of SL2 on C[x0 , . . . , x3 ]4 is as follows: C[x0 ,...,x2 ]4

⊕ C[x0 ,...,x2 ]3 ·x3 ⊕ C[x0 ,...,x2 ]2 ·x32 ⊕ C[x0 ,...,x2 ]1 ·x33 ⊕ C·x34

V (8)⊕V (4)⊕V (0)

V (6)⊕V (2)

V (4)⊕V (0)

V (2)

V (0)

(42)

Now let us determine the sub-representation Ua,b containing [fa,b ] and such that Hom([fa,b ], Ua,b /[fa,b ]) is the tangent space at V (fa,b ) to the orbit PGL(4)V (fa,b ) (we only assume that (a, b) = (0, 0)). Let i ∈ C[x0 , . . . , x3 ]1 for i ∈ {0, . . . , 3}; we will write out the term multiplying t in the expansion of fa,b (x0 + t0 , . . . , x3 + t3 ) as element of C[x0, . . . , x3 ]4 [t] for various choices of i ’s. For i = μi x3 , we get ! 4q

2 

" μi x i

x3 + 2(a + b)μ3qx32 + 2(a + b)

i=0

! 2 

" μi xi x33 + 4abμ3x34 .

(43)

i=0

Letting i ∈ C[x0 , x1 , x2 ]1 , we get 4q

! 2 

" i x i

+ 2(a + b)q3x3 + 2(a + b)

i=0

! 2 

" i xi x32 + 4ab3x33 .

(44)

i=0

It follows that Ua,b

2 V (4) ⊕ V (2)2 ⊕ V (0)2 ∼ = V (4) ⊕ V (2) ⊕ V (0)2

if a = b, if a = b.

(45)

The difference between the two cases is due to the different behaviour of the V (2)-representations appearing in (43), (44) and contained in the direct sum C[x0 , . . . , x2 ]3 · x3 ⊕ C[x0 , . . . , x2 ]1 · x33 . If a = b, the representations in (43) and (44) are distinct, if a = b they are equal.  at Points of W 1 \ Eω 5.4.2 The Germ of M  → M is an isomorphism away from {ω, υ}. Since W1 does not The map φ : M  at a point  1 \ Eω ) is identified by φ with the germ contain υ, the germ of M x ∈ (W of M at x := φ( x ). Let us examine the germ of M at a point x ∈ (W1 \ {ω}). There exists (a, b) ∈ C2 , with a = b, such that a polystable quartic representing x is V (fa,b ), where fa,b is as in (41). Keeping notation as in Sect. 5.4.1, SL2 acts on

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V (fa,b ). Let Na,b ⊂ C[x0, . . . , x3 ]4 be the sub SL2 -representation Na,b := V (8) ⊕ V (6) ⊕ R · x32 ⊕ *2qx32 + (a + b)x34,,

(46)

where R ⊂ C[x0 , x1 , x2 ]2 is the summand isomorphic to V (4), and let Na,b := {V (fa,b + g) | g ∈ Na,b }.

(47)

Proposition 10 Keeping notation as above, Na,b is an AutV (fa,b )-invariant normal slice to the orbit PGL(4)V (fa,b ). Proof Let Ua,b ⊂ C[x0 , . . . , x3 ]4 be as in Sect. 5.4.1; thus P(Ua,b ) is the projective tangent space at V (fa,b ) to the orbit PGL(4)V (fa,b ). Then Ua,b is the sum of the two SL2 -representations in (43) and (44) (and, as representation, it is given by the first case in (45)), and it follows that the SL2 -invariant affine space in (47) is transversal to P(Ua,b ) at V (fa,b ). Lastly, Na,b is AutV (fa,b )-invariant because AutV (fa,b ) is generated by the image of SL2 and the reflection in the plane x3 = 0. The natural map ψ : Na,b //AutV (fa,b ) −→ M

(48)

is étale at V (fa,b ) by Luna’s étale slice Theorem. For later use, we make the following observation. Claim Keep notation and assumptions as above, in particular a = b. Let η : Na,b → M be the composition of the quotient map Na,b → Na,b //AutV (fa,b ) and the map ψ in (48). Then η({V (fa,b + t (2qx32 + (a + b)x34 )) | t ∈ C}) ⊂ W1 .

(49)

Moreover, let U ⊂ Na,b be an AutV (fa,b )-invariant open (in the classical topology) neighborhood of fa,b such that the restriction of ψ to U //AutV (fa,b ) is an isomorphism onto ψ(U //AutV (fa,b )); then x ∈ U is mapped to W1 by η and has closed SL2 -orbit if and only if x = V (fa,b + t (2qx32 + (a + b)x34 )) for some t ∈ C. Proof The first statement follows from a direct computation. In fact, an easy argument shows that there exist holomorphic functions ϕ, ψ of the complex variable t vanishing at t = 0, such that (q + (a + ϕ(t))x32 ) · (q + (b + ψ(t))x32 ) = fa,b + t (2qx32 + (a + b)x34 ). The second statement holds because W1 is an irreducible curve, and so is the lefthand side of (49).

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 at the Unique Point in W 1 ∩ Eω 5.4.3 The Germ of M Let Mω → M be the blow-up of (the reduced) ω. We may work on Mω , since W1 does not contain υ. Let P → |OP3 (4)| be the blow-up with center the closed subset D parametrizing double quadrics, and let ED be the exceptional divisor. By Kirwan [24] the blow-up Mω is identified with the quotient of P by the natural action of PGL(4) (with a polarization close to the pull-back of the hyperplane line-bundle on |OP3 (4)|)—see the proof of Proposition 8). We will describe an SL2 -invariant normal slice in P to the PGL(4)-orbit of a point representing the unique point in 1 ∩ Eω . First, recall that we have an identification Eω = Mh , where Mh is the W 1 ∩ GIT hyperelliptic moduli space in (36), see Proposition 8. The unique point in W 3 Eω is represented by a point in ED mapping to a smooth quadric Q ⊂ P , and corresponding to 4 ∈ P(H 0(OQ (4))) (recall that the fiber of the exceptional divisor over Q is identified with P(H 0 (OQ (4)))), where 0 =  ∈ H 0 (OQ (1)) is a section with smooth zero-locus (a smooth conic); moreover the points we have described have closed orbit in the locus of PGL(4)-semistable points. 1 ∩ Eω by the point with closed orbit Remark 18 We represent the unique point in W (V (q + ax32 ), x34 ) ∈ ED (notation as above), where q is as in Sect. 5.4.1 and a = 0. In order to simplify notation, we let Qa := V (q + ax32 ), and p := (Qa , x34 ) ∈ ED . Now let S ⊂ C[x0 , . . . , x3 ]4 be the sub SL2 -representation S := V (8) ⊕ V (6) ⊕ R · x32 ⊕ *x34 ,,

(50)

where R ⊂ C[x0 , x1 , x2 ]2 is the summand isomorphic to V (4). (Notice the similarity with (46).) Let Sa := {V (fa,a + g) | g ∈ S}

(51)

Claim Keeping notation as above, the double quadric V (fa,a ) is an isolated and reduced point of the scheme-theoretic intersection between the affine space Sa and the closed D ⊂ |OP3 (4)| parametrizing double quadrics. Proof Of course V (fa,a ) ∈ D, because fa,a = (q + ax32)2 . Let TV (fa,a ) Sa and TV (fa,a ) D be the tangent spaces to Sa and D at V (fa,a ) respectively; we must show that their intersection (as subspaces of TV (fa,a ) |OP3 (4)|) is trivial. We have TV (fa,a ) Sa = Hom(*fa,a ,, *S, fa,a ,/*fa,a ,),

TV (fa,a ) D = Hom(*fa,a ,, Ua,a /*fa,a ,),

where Ua,a is as in Sect. 5.4.1. As is easily checked, S ∩ Ua,a = {0}. Thus *S, fa,a , ∩ Ua,a = *fa,a ,, and the claim follows.

(52)

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By Claim 5.4.3 the scheme-theoretic intersection D ∩ Sa is the disjoint union of the reduced singleton {V (fa,a )} and a subscheme Ya . Let Ua := Sa \ Ya ; then Ua is an open neighborhood of V (fa,a ) in Sa , and it is invariant under the action of AutV (fa,a ). Let  Ua ⊂ P be the strict transform of Ua (recall that P → |OP3 (4)| is the blow-up with center D), and let ϕ :  Ua → Ua be the restriction of the contraction P → |OP3 (4)|. By Claim 5.4.3 ϕ is the blow-up of the (reduced) point V (fa,a ). Remark 19 Since fa,a , x34 ∈ Sa , the point p = (Qa , x34 ) ∈ ED (see Remark 18) belongs to  Ua . Moreover the stabilizer (in PGL(4)) of p is equal to O(q) i.e. to AutV (fa,b ) for a = b (see Sect. 5.4.1), and it preserves  Ua . Proposition 11 Keeping notation as above,  Ua is a Stab(p)-invariant normal slice to the orbit PGL(4)p in P . Proof Let Y := PGL(4)p. We must prove that the tangent space to  Ua at p is transversal to the tangent space to Y at p. First notice that dim Y = 12 and dim  Ua = 22, hence dim Y + dim  Ua = dim P . Thus it suffices to prove that Tp Y ∩ Tp  Ua = {0}.

(53)

Let π : P → |OP3 (4)| be the blow up of D. By Claim 5.4.3, dπ(p)(Tp  Ua ) = Hom(*fa,a ,, *fa,a , x34 ,/*fa,a ,). On the other hand, dπ(p)(Tp Y ) = Tπ(p)D, and hence dπ(p)(Tp  Ua ) ∩ dπ(p)(Tp Y ) = {0}. It follows that the intersection on the left hand side of (53) is contained in the kernel of the restriction of dπ(p) to Tp  Ua ∩ Eπ(p)), Ua , i.e. Tp ( where Eπ(p) is the fiber of ED → D over π(p) = V (fa,a ). Hence it suffices to prove that Ua ∩ Eπ(p)) = {0}. Tp Y ∩ Tp (

(54)

The fiber Eπ(p) is naturally identified with PH 0 (OQa (4)). With this identification, we have Tp Y ∩ Tp Eπ(p) = Hom(*x34 ,, C[x0 , . . . , x3 ]1 · x33 /*x34 ,), Tp ( Ua ∩ Eπ(p)) = Hom(*x34 ,, S/*x34 ,). Here we are abusing notation: C[x0 , . . . , x3 ]1 · x33 and S stand for their images in H 0 (OQa (4)). Since the kernel of the restriction map H 0 (OP3 (4)) → H 0 (OQa (4)) is equal to Ua,a , Eq. (54) follows from the equalities *C[x0 , . . . , x3 ]1 · x33 , S, ∩ Ua,a = {0}, (C[x0 , . . . , x3 ]1 · x33 ) ∩ S = *x34 ,

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The natural map ψ:  Ua //Stab(p) −→ M

(55)

is étale at p by Luna’s étale slice Theorem. The result below is the analogue of Claim 5.4.2. Claim Keep notation and assumptions as above. Let ζ :  Ua → M be the composition of the quotient map  Ua →  Ua //Stab(p) and the map ψ in (55). Let C ⊂  Ua be the strict transform of the line {V (fa,a + tx34 ) | t ∈ C}. Then ζ(C) ⊂ W1 . Moreover, let U ⊂  Ua be a Stab(p)-invariant open (in the classical topology) neighborhood of p such that the restriction of ψ to U //StabV (p) is an isomorphism onto ψ(U //Stab(p)); then x ∈ U is mapped to W1 by ζ and has closed SL2 -orbit if and only if x = V (fa,a + tx34 ) for some t ∈ C. Proof First (q + (a + u)x32 )(q + (a − u)x32 ) = fa,a − u2 x34 shows that ζ(C) ⊂ W1 . For the remaining statement see the proof of Claim 5.4.2. 5.4.4 Moduli of K3 Surfaces Which Are Generic Double Cones Let Λ be the graded C-algebra Λ := S• (V (4)∨ ⊕ V (6)∨ ⊕ V (8)∨ ),

(56)

where V (2d)∨ has degree d. Then PSL(2) acts on Proj Λ, and OProjΩ (1) is naturally linearized. The involution Proj Λ −→ Proj Λ [f, g, h] → [f, −g, h] commutes with the action of PSL(2), and hence there is a (faithful) action of Gc := PSL(2) × Z/(2)

(57)

Mc := Proj Λ//Gc

(58)

on Proj Λ. We let

be the GIT quotient. We will show that Mc is naturally a compactification of the moduli space of hyperelliptic quartic K3 surfaces which are double covers of a quadric cone with branch divisor not containing the vertex of the cone. First, we think of SL2 as the double cover of SO(q), where q = x02 + x12 + x22 is as in Sect. 5.4.2, and correspondingly V (2d) is a subrepresentation of C[x0 , x1 , x2 ]d .

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We associate to ξ := (f, g, h) ∈ V (4) ⊕ V (6) ⊕ V (8), the quartic Bξ := V (x34 + f x32 + gx3 + h).

(59)

Thus V (4) ⊕ V (6) ⊕ V (8) is identified with the set of such quartics. Both Gc and the multiplicative group C∗ act on the set of such quartics (the second group acts by rescaling x3 ). The quotient of (V (4) ⊕ V (6) ⊕ V (8)) \ {0} by the C∗ action is Proj Λ, hence Mc is identified with the quotient (V (4) ⊕ V (6) ⊕ V (8)) \ {0} by the full Gc × C∗ -action. Given [ξ ] ∈ Proj Λ, we let Xξ be the double cover of the cone V (q) ⊂ P3C ramified over the restriction of Bξ to V (q), and Lξ be the degree-4 polarization of Xξ pulled back from OP3 (1). Proposition 12 Let [ξ ] ∈ Proj Λ be such that Xξ has rational singularities. Then [ξ ] is Gc -stable. The open dense subset of Mc parametrizing isomorphism classes of such [ξ ] is the moduli space of polarized quartics which are double covers of a quadric cone with branch divisor not containing the vertex of the cone. Proof Let [ξ ] = [f, g, h] ∈ Proj Λ be a non-stable point. Then by the HilbertMumford Criterion there exist a point a ∈ P1 (where P1 is identified with the conic V (q, x3 ) via the Veronese embedding) such that multa (f ) ≥ 2,

multa (g) ≥ 3,

multa (h) ≥ 4.

(60)

The point a ∈ P1 is identified with a point p ∈ V (q, x3 ) (as recalled above), which belongs to the quartic Bξ . The inequalities in (60) give that the multiplicity at p of the divisor Bξ |V (q) is at least 4, and hence the corresponding double cover of V (q) (i.e. Xξ ) does not have rational singularities. This proves the first statement. The rest of the proof is analogous to Shah’s proof (see Theorem 4.3 in [50]) that Mu (see (26)) is a compactification of the moduli space for unigonal K3 surfaces. The key point is that any quartic not containing the vertex [0, 0, 0, 1] has such an equation after a suitable projectivity ϕ (a Tschirnhaus transformation) of the form ϕ ∗ xi = xi , ϕ ∗ x3 = x3 + (x0 , x1 , x2 ) where (x0 , x1 , x2 ) is homogeneous of degree 1. Let [ξ ] ∈ Proj Λ be generic; then (Xξ , Lξ ) is a polarized quartic K3 surface whose (2) period point belongs to Hh , which (see [34]) is identified with F (17) via the embedding f17,19 : F (17) → F . Thus we have a rational period map pc : Mc  F (17)∗ ⊂ F ∗ .

(61)

A generic polarized quartic K3 surface is a double cover of the quadric cone unramified over the vertex, and hence is isomorphic to (Xξ , Lξ ) for a certain [ξ ] ∈ Proj Λ. By the global Torelli Theorem for K3 surfaces, it follows that the period map pc is birational.

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5.4.5 Partial Extension of the Period Map on a Weighted Blow-Up: 1 \ Eω The Case of a Point in W Let (a, b) ∈ C2 , with a = b. Let Na,b be the SL2 representation in (46), and let Ma,b be the sub-representation Ma,b := V (8) ⊕ V (6) ⊕ R · x32 .

(62)

Let Na,b be the normal slice of V (fa,b ) defined in Sect. 5.4.2, and let Ma,b ⊂ Na,b be the subspace Ma,b := {V (fa,b + g) | g ∈ Ma,b }. Notice that dim Ma,b = 21. Let (z1 , . . . , z5 ) be coordinates on V (4), let (z6 , . . . , z12 ) be coordinates on V (6), and let (z13 , . . . , z21 ) be coordinates on V (8); thus (z1 , . . . , z21 ) are coordinates on Ma,b (with a slight abuse of notation) centered at V (fa,b ). Let σ be the weight defined by ⎧ ⎪ ⎪ ⎨2 if i ∈ {1, . . . , 5}, (63) σ (zi ) := 3 if i ∈ {6, . . . , 12}, ⎪ ⎪ ⎩4 if i ∈ {13, . . . , 21}. a,b := Blσ (Ma,b ) be the corresponding weighted blow up, and let Ea,b be Let M a,b → Ma,b . Thus Ea,b is the weighted projective space the exceptional set of M 5 7 9 ∼ P(2 , 3 , 4 ) = Proj Λ, where Λ is the graded ring in (56) (with grading defined right after (56)). The action of Aut(Vfa,b ) = Gc (here Gc is as in (57)) on Ma,b lifts a,b . Thus there is an associated GIT quotient M a,b //Gc . The map to an action on M a,b → Ma,b induces a map M a,b //Gc −→ Ma,b //Gc . θ: M

(64)

Moreover, we have the set-theoretic equality θ −1 (V (fa,b ))red = Proj Λ//Gc = Mc .

(65)

Since the natural map Ma,b //Gc → M is dominant, it makes sense to compose it with the (rational) period map p : M  F ∗ . Composing with the birational map in (64), we get a rational map a,b //Gc  F ∗ . pa,b : M

(66)

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Proposition 13 With notation as above, the restriction of pa,b to θ −1 (V (fa,b ))red = Mc is equal to the composition of the automorphism ϕa,b

Mc −→ Mc i [f, g, h] → [f, − 2 (a − b)g, − 14 (a − b)2 h]

(67)

and the period map in (61). Moreover pa,b is regular at all points of θ −1 (V (fa,b ))red where pc is regular. Proof Let [ξ ] = [f, g, h] ∈ Proj Λ = Ea,b be a Gc -semistable point with corresponding point [ξ ] ∈ Mc , and let [η] = ϕa,b ([ξ ]). Suppose that the period map pc is regular at [η]. We will prove that if Δ ⊂ C is a disc centered at 0, and a,b is an analytic map mapping 0 to [ξ ] and no other point to the exceptional Δ→M divisor Ea,b , then the period map is defined on a neighborhood of 0 ∈ Δ, and its value at 0 is equal to the period point of (Xη , Lη ). This will prove the Proposition, by Corollary 1. By Proposition 7 the statement that we just gave boils down to the following computation. First, we identify V (2d) with the corresponding SO(q)-subrepresentation of C[x0 , x1 , x2 ]d ; thus f, g, h ∈ C[x0 , x1 , x2 ] are homogeneous of degrees 2, 3 and 4 respectively. Now let X ⊂ P3 × Δ be the hypersurface given by the equation 0 = (q + ax32 )(q + bx32) + t 2 x32 (f + tF ) + t 3 x3 (g + tG) + t 4 (h + tH )

(68)

where F ∈ C[x0 , x1 , x2 ]2 [[t]], G ∈ C[x0 , x1 , x2 ]3 [[t]], and H ∈ C[x0 , x1 , x2 ]4 [[t]]. Now consider the 1-parameter subgroup of GL4 (C) defined by λ(t) := diag(1, 1, 1, t). We let Y ⊂ P3 × Δ be the closure of {([x], t) | t = 0,

λ(t)[x] ∈ X }.

Then Yt ∼ = Xt for t = 0, and Y has equation 0 = q 2 + t 2 (a + b)x32q + t 4 (abx34 + x32 f + x3 g + h) + t 5 (. . .)

(69)

Let ν : Y → Y be the normalization of Y . Dividing (69) by t 4 xi4 , we get that the ring of regular functions of the affine set ν −1 (Y ∩P3xi ) is generated over C[Y ∩P3xi ] by the rational function ξi := q/(xi2 t 2 ), which satisfies the equation %

0=

ξi2

x3 + (a + b) xi

&2 ξi + (abx34 + x32 f + x3 g + h)/xi4 + t (. . .)

It follows that for t → 0 the quartics Xt approach the double cover of V (q) branched over the intersection with the quartic 0 = ((a + b)x32)2 − 4(abx34 + x32 f + x3 g + h) = (a − b)2 x34 − 4x32f − 4x3 g − 4h. (70)

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5.4.6 Partial Extension of the Period Map on a Weighted Blow-Up: 1 ∩ Eω The Unique Point in W Let a = 0, and  Ua ∩ E D . Va :=  (We recall that ED is the exceptional divisor of the blow-up P → |OP3 (4)| with center the closed subset D parametrizing double quadrics.) Thus, letting S be as in (50), we have  Va = P(S) = P(V (8) ⊕ V (6) ⊕ R · x32 ⊕ *x34 ,),

dim  Va = 21.

(71)

Va , see Remark 18. Then  Va is mapped to itself by Stab(p), Let p := (Qa , x34 ) ∈  and by restriction of the map ψ in (55) we get a map   Va //Stab(p) −→ M. We define a weighted blow up of  Va with center p as follows. First, by (71) we have the following description of an affine neighborhood T of p ∈  Va : V (8) ⊕ V (6) ⊕ R · x32 −→ T α → [x34 + α] Let (z1 , . . . , z5 ) be coordinates on R · x32 = V (4), let (z6 , . . . , z12 ) be coordinates on V (6), and let (z13 , . . . , z21 ) be coordinates on V (8); thus (z1 , . . . , z21 ) are coordinates on T (with a slight abuse of notation) centered at the point p. Let σ be the weight defined by ⎧ ⎪ ⎪ ⎨2 σ (zi ) := 3 ⎪ ⎪ ⎩4

if i ∈ {1, . . . , 5}, if i ∈ {6, . . . , 12},

(72)

if i ∈ {13, . . . , 21}.

(Note: we are proceeding exactly as in Sect. 5.4.5.) Let Va := Blσ ( Va ) be the corresponding weighted blow up, and let Ea be the corresponding exceptional divisor. Thus Ea is the weighted projective space P(25, 37 , 49 ) ∼ = Proj Λ, where Λ is the graded ring in (56) (with grading defined right after (56)). The action of Aut(p) on  Va lifts to an action on Va There is an associated GIT quotient Va //Stab(p), and a regular map η: Va //Stab(p) −→  Va //Stab(p).

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We have the set-theoretic equality η−1 (p)red = Proj Λ//Gc = Mc .

(73)

Va //Aut(p)  F ∗ . pa :

(74)

We have a rational map

Proposition 14 With notation as above, the restriction of pa to η−1 (p)red = Mc is equal to the period map in (61). Moreover pa is regular at all points of  η−1 (p)red where pc is regular. Proof Let [ξ ] = [f, g, h] ∈ Proj Λ = Ea be a Gc -semistable point with corresponding point η ∈ Mc . Suppose that the period map pc is regular at η. We will prove that if Δ ⊂ C is a disc centered at 0, and Δ → Va is an analytic map mapping 0 to [ξ ] and no other point to the exceptional divisor Ea , then the period map is defined on a neighborhood of 0 ∈ Δ, and its value at 0 is equal to the period point of (Xη , Lη ). This will prove the Proposition, by Corollary 1. By Proposition 7, the previous statement boils down to the following computation. Let f, g, h ∈ C[x0 , x1 , x2 ] be homogeneous of degrees 2, 3 and 4 respectively, not all zero. Let Ct ⊂ V (q + ax32) be the intersection with the quartic x34 + t 2 x32 (f + tF ) + t 3 x3 (g + tG) + t 4 (h + tH ) = 0. where F ∈ C[x0 , x1 , x2 ]2 [[t]], G ∈ C[x0 , x1 , x2 ]3 [[t]], and H ∈ C[x0 , x1 , x2 ]4 [[t]]. We will show that Ct for t = 0 approaches for t → 0, the curve q = x34 + x32 f + x3 g + h = 0. In fact it suffices to consider the limit for t → 0 of λ(t)Ct , where λ is the 1-PS λ(t) = (1, 1, 1, t).  and Partial Extension of the Period 5.4.7 A Global Modification of M Map Let T ⊂ |OP3 (4)| be the closure of the set of PGL(4)-translates of V (fa,b ), for all (a, b) ∈ C2 . Thus T is a closed, PGL(4)-invariant subset, containing D (the set of double quadrics), and dim T = 13.

(75)

Let  T ⊂ P be the strict transform of T in the blow-up π : P → |OP3 (4)| with center D. The set of semistable points  Tss ⊂  T (for a polarization π ∗ L (−ED ) ∗ close to π L , see the proof of Proposition 8) is the union of the set of points of

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 T \ ED which are mapped by π to quartics PGL(4)-equivalent to V (fa,b ) for some ss a = b, and of  T ∩ ED . The latter set consists of the PGL(4)-translates of the points 4 (Qa , x3 ) defined in Remark 18. In Sects. 5.4.5 and 5.4.6 we defined a weighted blow up of an explicit normal slice to  T at points x ∈  Tss . That construction can be globalized: one obtains a modification π : P → P which is an isomorphism away from P \  T, and replaces ss  T by a locally trivial fiber bundle over  Tss with fiber isomorphic to the weighted projective space P(25 , 37 , 49 ). In fact the weighted blow up is isomorphic to the usual blow up of a suitable ideal, see Remark 2.5 of [3], hence one may define an ideal I co-supported on  T such that P = BlI P . Let E π . Letting LP := π ∗ L (−ED ) be a T be the exceptional divisor of polarization of P as above, we may consider the GIT quotient of P with PGL(4) linearized polarization LP := π ∗ LP (−tE T ), call it M(t). For 0 < t small enough,  the map π induces a regular map M(t) → M. From now on we drop the parameter denotes M(t) t from our notation; thus M for t small. The image of E T in M is a fiber bundle 1 , ρ : E1 → W  F ∗ be the period map. We p: M with fiber Mc over every point. Let 1 over x is regular away claim that the restriction of p to the fiber of E1 → W ∗ from the indeterminacy locus of pc : Mc  F , and it has the same value, provided we compose with the automorphism of Mc given by (67) if x ∈ / Eω and π (x) = [V (fa,b )]. In order to prove the claim it suffices to prove the following. Let Δ ⊂ C be a disc be an analytic map mapping 0 to a point centered at 0, and let Δ → M x ∈ E1 such that the period map pc is regular at the point η ∈ Mc = ρ −1 (ρ( x )) corresponding to x , and suppose that (Δ \ {0}) is mapped to the complement of E1 and into the locus where the period map is regular; then the value at 0 of the extension of the period map on Δ \ {0} is equal to the period point of (Xη , Lη ). We may assume that lifts to an analytic map τ : Δ → P mapping 0 to a point of E  with closed Δ→M T orbit (in the semistable locus) lifting x . In Sects. 5.4.5 and 5.4.6 we have checked that the value at 0 of the extension behaves as required if π ◦τ (Δ) is contained in the normal slice to T at the point π ◦τ (0) (defined in Sects. 5.4.5 and 5.4.6 respectively). It remains to prove that it behaves as required also if the latter condition does not hold. If π ◦ τ (0) ∈ / ED , then the argument is similar to that given in Sect. 5.4.5; one simply replaces a, b ∈ C by holomorphic functions a(t), b(t) where t ∈ Δ. If π ◦ τ (0) ∈ ED , one needs a separate argument. The relevant computation goes as follows. Let f, g, h ∈ C[x0 , x1 , x2 ] be homogeneous of degrees 2, 3 and 4 respectively, not all zero. Let X ⊂ P3 × Δ be the hypersurface given by the equation (q + x32 )2 + t 4k x34 + t 4k+6p x32 (f + tF ) + t 4k+9p x3 (g + tG) + t 4k+12p (h + tH ) = 0, (76)

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where F ∈ C[x0 , x1 , x2 ]2 [[t]],

G ∈ C[x0 , x1 , x2 ]3 [[t]],

H ∈ C[x0 , x1 , x2 ]4 [[t]].

Let λ(t) := diag(1, 1, 1, t 4 ), and let Y ⊂ P3 × Δ be the closure of {([x], t) | t = 0,

λ(t)[x] ∈ X }.

Thus Yt ∼ = Xt for t = 0, and Y has equation q 2 + 2t 8 qx32 + t 16 x34 + t 4k+16 x34 + t 4k+6p+8 x32 (f + tF ) + t 4k+9p+4 x3 (g + tG)+ t 4k+12p (h + tH ) = 0. (77) Dividing the above equation by t 16 we find that the rational function ξi := q/(xi2t 8 ) satisfies the equation % ξi2 + 2

x3 xi

&2 ξi + (x34 + t 4k x34 + t 4k+6p−8 x32 (f + tF ) + t 4k+9p−12 x3 (g + tG)+ t 4k+12p−16(h + tH ))/xi4 = 0.

It follows that the fiber at t = 0 of the normalization of Y is the double cover of V (q) ramified over the intersection with the limit for t → 0 of the quartic 4x34 −4(x34 +t 4k x34 +t 4k+6p−8 x32 (f +tF )+t 4k+9p−12 x3 (g+tG)+t 4k+12p−16 (h+tH )) = 0.

Replacing x3 by t −3p+4 x3 we get that the fiber at t = 0 of the normalization of Y is the double cover of V (q) ramified over the intersection with the quartic x34 + x32 f + x3 g + h = 0.

(78)

Let us explain why the above computation proves the required statement. Let  : Δ → |OP3 (4)| be the analytic map defined by (t) := Xt . Then Im() ⊂ S1 , 1 be the blow up of S1 \ Y1 with where S1 is as in (51) (notice that X0 = f1,1 ). Let U  center V (f1,1 ), see Sect. 5.4.3, and let   : Δ → U1 be the lift of  (by shrinking Δ we may assume that Im() ∩ Y1 = ∅). Then   (0) = p = (Q1 , x34 ), notation as in Remark 18. Now, choose a basis {a0, . . . , a21 } of the SL2 -representation S given by (50) adapted to the decomposition in (50); more precisely a0 = x34 , {a1 , . . . , a5 } is a basis of R · x32 , {a6 , . . . , a12 } is a basis of V (6), and {a9 , . . . , a21 } is a basis of V (8). Let {w0 , . . . , w21 } be the basis dual to {a0 , . . . , a21 }; then (w0 , . . . , w21 ) are coordinates on an affine neighborhood of V (fa,a ) in Sa , centered at V (fa,a ). Next set y0 = w0 , and yi = wi /w0 for i ∈ {1, . . . , 22}. Then (y0 , . . . , y21 ) are coordinates on an affine

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neighborhood of p ∈  U1 , centered at p. Let (z1 , . . . , z21 ) be the affine coordinates introduced in Sect. 5.4.6; we may assume that yi | V1 = zi for i ∈ {1, . . . , 22}. In the coordinates (y0 , . . . , y21 ) we have  (t)=(t 4k ,t 6p (f1 +tF1 ),...,t 6p (f5 +tF5 ),t 9p (g5 +tG5 ),...,t 9p (g12 +tG12 ),t 12p (h13 +tH13 ),...,t 12p (h21 +tH21 )),

with obvious notation: (f1 , . . . , f5 ) are the coordinates of f in the basis {a1 , . . . , a5 }, etc. The computation above shows that the extension at 0 of the period map is equal to the period point of the double cover of V (q) ramified over the intersection with the quartic defined by (78), and hence the period map is regular at the point corresponding to [f, g, h] by Proposition 7 and Corollary 1.  5.4.8 The First Flip and a Contraction of M is isomorphic to W 1 × Mc . The normal bundle of E1 restricted The divisor E1 ⊂ M to the fibers of the projection E1 → Mc is negative; it follows that (in the analytic → M1/2 of E1 along the fibers of E1 → Mc . category) there exists a contraction M  F We claim that M1/2 must be isomorphic to F (1/3, 1/2). In fact, let p: M be the period map (notice: contrary to previous notation, the codomain is F , not F ∗ ). The generic fiber of E1 → Mc is in the regular locus of p, and is mapped to a constant: it follows that 1 × {[f, g, h]}) = 1 × {[f, g, h]}), 0 = p∗ (λ) · (W p∗ (Δ) · (W

[f, g, h] ∈ Mc . (79)

1 , and adopting the notation of [34], we have On the other hand, letting p ∈ W p({p} × Mc,reg ) ⊂ Im(f17,19 ).

(80)

(Here Mc,reg is the set of regular points of the period map pc : Mc  F ; by Proposition 14 it is equal to the intersection of {p} × Mc,reg with the set of ∗ regular points of p.) By Proposition 5.3.7 of [34] we have f17,19 (λ + βΔ) = (1 − 2β)λ(17) + βΔ(17). Now, Δ(17) = Hh (17)/2, and p({p} × Mc ) avoids the support of Hh (17) = Im f16,17 . Thus p∗ (λ + βΔ)|{p}×Mc = p∗ ((1 − 2β)λ)|{p}×Mc .

(81)

The conclusion is that p∗ (λ + βΔ) contracts all of E1 to a point if β ≥ 1/2 (and is trivial on E1 if β = 1/2), while if β < 1/2, then the restriction of p∗ (λ + βΔ) to E1 is the pull-back of an ample line bundle on Mc . Thus we expect that for β < 1/2 close to 1/2 the (Q) line-bundle p∗ (λ + βΔ) is the pull-back of an ample (Q) line bundle on M1/2 , and hence M1/2 is identified with F (β), because the period map

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would be birational map which is an isomorphism in codimension 2 and pulls back an ample line bundle to an ample line bundle.

6 Semistable Reduction for Dolgachev Singularities, and the Last Three Flips In the present section, we will provide evidence in favour of the predictions that there are flips corresponding to β ∈ { 16 , 17 , 19 } (the critical values of β closest to β = 0, which corresponds to F ∗ ), with centers birational to the loci of quartics with E14 , E13 , and E12 singularities respectively. There is a strong similarity with the first steps in the Hassett-Keel program. Specifically, in the variation of log canonical models Mg (α) = Proj(M g , KM g + αΔM g ) (for α ∈ [0, 1]) for the moduli space 9 of genus g curves M g , the first critical value is α = 11 which corresponds to replacing the curves with elliptic tails by cuspidal curves. Similarly, at the next 7 critical value α = 10 , the locus of curves with elliptic bridges is replaced by the locus of curves with tacnodes (see [19, 20] for details). In the proposed analogy, the singularities E12 , E13 , and E14 (the simplest 2 dimensional non-log canonical singularities) correspond to cusps and tacnodes, while, as we will see, certain lattice polarized K3 surfaces correspond to elliptic tails and bridges.

6.1 KSBA (Semi)Stable Replacement According to the general KSBA philosophy, for varieties of general type there exists a canonical compactification obtained by allowing degenerations with semilog-canonical (slc) singularities and ample canonical bundle. In particular, any 1parameter degeneration has a canonical limit with slc singularities. However, when studying GIT one ends up with compactifications that allow non-slc singularities. For example, the GIT compactification for quartic curves will allow quartics with cusp singularities. Thus a natural question is: given a degenerations X /Δ of varieties of general type such that the general fiber is smooth (or mildly singular), but such that X0 does not have slc singularities, to find a stable KSBA replacement X0 . Of course, X0 depends on the original fiber X0 and on the family X /Δ (i.e. the choice of the curve in the moduli space with limit X0 ). Motivated by the Hassett-Keel program, Hassett [17] studied the influence of certain classes of curve singularities on the KSBA (semi)stable replacement (in this case, the usual nodal curve replacement). Hassett’s perspective is to consider a curve C0 with a unique non-slc (i.e. non-nodal) singularity, and to examine C /Δ, a generic smoothing of C0 . The question is what can be said about the semi-stable replacement C0 of C0 . 0 of C0 (assuming Of course, one component of C0 will be the normalization C that this normalization is not a rational curve). The remaining components (and the 0 ) of C (the “tail part”) will depend on the non-slc singularity of C0 and gluing to C 0

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its smoothing; one determines them by a local computation. The classical example is the semi-stable replacement for curves with an ordinary cusp (see [16, §3.C]), that we briefly review below. Example 1 (Semi-Stable Replacement for Cuspidal Curves) Locally (in the analytic topology) a curve in a neighborhood of an ordinary cusp has equation y 2 +x 3 = 0, and a generic 1-parameter smoothing will be given by C := V (t +y 2 +x 3) → Δt . After a base change of order 6, which is necessary to make the local monodromy action unipotent, one obtains a surface V (t 6 + y 2 + x 3 ) ⊂ (C3 , 0) with a simple elliptic singularity at the origin. The weighted blow-up of the origin will resolve this singularity, and the resulting exceptional curve E is an elliptic curve (explicitly it is V (t 6 + y 2 + x 3 ) ⊂ W P(1, 3, 2)). The new family C (obtained by base change and weighted blow-up) will be a semi-stable family of curves, with the new central fiber consisting of the union of the normalization of C0 and of the exceptional curve E (“the elliptic tail”) glued at a single point. Note that instead of a weighted blow-up, one can use several regular blow-ups, these will lead first to a semi-stable curve with additional rational tails, which can be then contracted to give the stable model (with a single elliptic tail). The two blow-up (and then blow-down) processes are equivalent; the weighted blow-up has the advantage of being minimal, and it generalizes well in our situation. As mentioned above, Hassett [17] has generalized this for certain types of planar curve singularities (essentially weighted homogeneous, and related). In higher dimension (e.g. surfaces), much less is known—there is a similar computation (for surfaces with triangle singularities) to the elliptic curve example contained in an unpublished letter of Shepherd-Barron to Friedman (in connection to [9]— such examples tend to give degenerations with finite, or even trivial, monodromy). Similar computations appear in [12], and what is needed for our purposes will be reviewed below. Of course, we are concerned with degenerations of K3 surfaces, thus the KSBA replacement strictly speaking doesn’t make sense (the main issue is non-uniqueness of the replacement). Nonetheless, given a degeneration X ∗ /Δ∗ with general fiber a K3, there exists a filling with X0 being a surface with slc singularities (and trivial dualizing sheaf). This follows from the Kulikov-Person-Pinkham theorem and Shepherd-Barron [52, 53]. Furthermore, if X0 has a unique non-log canonical singularity, we can ask (mimicking Hassett [17]): What is the KSBA replacement for 0 a quartic surface X0 with a single E12 singularity? In this case the resolution X is rational (this is analogous to the fact that the normalization of a cuspidal cubic curve is rational), and thus the focus is on the “tail” part.

6.2 Dolgachev Singularities The singularities that interests us are particular cases of Dolgachev singularities [5] (aka triangle singularities or exceptional unimodal singularities, the latter is the

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terminology used by Arnold et al. [4]). They are arguably the simplest 2 dimensional non-log canonical singularities, for this reason we view them as analogues of 1 dimensional ordinary cusps. Dolgachev singularities are hypersurface singularities with the property that they have a (non-minimal) resolution with exceptional divisor E +E1 +E2 +E3 , where E 2 = −1, E12 = −p, E22 = −q, E32 = −r, and the curves Ei only meet E transversely (comb type picture). By contracting the Ei ’s, we obtain a partial resolution with a rational curve E going through 3 quotient singularities of types p1 (1, 1), q1 (1, 1) and 1r (1, 1). While any (p, q, r) (with p1 + q1 + 1r < 1) gives a non-log canonical surface singularity, only 14 choices of integers (p, q, r) lead to hypersurface singularities, these are the Dolgachev singularities. The Dolgachev numbers of the singularity are p, q, r. The cases relevant to us are E12 , E13 , and E14 , with Dolgachev numbers (2, 3, 7), (2, 4, 5), and (3, 3, 4) respectively. Remark 20 Very relevant in this discussion is the so called Tp,q,r graph (for p, q, r positive integers). This consists of a central node, together with 3 legs of lengths p − 1, q − 1, and r − 1 respectively. As usual to such a graph, one can associate an even lattice by giving a generator of norm −2 for each node, and two generators are orthogonal unless the corresponding nodes are joined by an edge in the graph (in which case, we define the intersection number to be 1). The cases p1 + q1 + 1r > 1 corresponding precisely to the ADE Dynkin graphs (with ADE associated lattices). For example (1, p, q) corresponds to Ap+q−1 , while (2, 3, 3) corresponds to E6 . Note also p1 + q1 + 1r > 1 is equivalent to the associated lattice being negative semi-definite. The three cases with p1 + q1 + 1r = 1 correspond to the extended r (r = 6, 7, 8), and in these cases the associated lattice Dynkin diagrams of type E is negative semi-definite. Finally, the cases with p1 + q1 + 1r < 1 lead to a hyperbolic lattice. that the absolute value of the discriminant will be  It is  easy to compute  1 1 1 pqr 1 − p + q + r . The lattice of vanishing cycles associated to a Dolgachev singularity is Tp ,q ,r ⊕ U for some integers (p , q , r ), which are called the Gabrielov numbers of the singularity. In particular, we note that p + q + r = (p + q + r − 2) + 2 = μ is the Milnor number of the singularities (i.e. the rank of the lattice of vanishing cycles is the Milnor number). In other words, associated to a Dolgachev singularity there are two triples of integers: the Dolgachev numbers (p, q, r) related to the resolution of the singularity, and the Gabrielov numbers (p , q , r ) related to the lattice of vanishing cycles (and the local monodromy associated with the singularity). In Table 2 below we give these numbers for the cases relevant to us. Arnold observed that the 14 Dolgachev singularities come in pairs of two with the property that the Table 2 The relevant Dolgachev singularities

Singularity E12 E13 E14

Dolgachev no. 2, 3, 7 2, 4, 5 3, 3, 4

Gabrielov no. 2, 3, 7 2, 3, 8 2, 3, 9

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Dolgachev and Gabrielov numbers are interchanged. This is part of the so called strange duality (see [8] for a survey). The key point is that Tp,q,r and Tp ,q ,r are mutually orthogonal in E82 ⊕ U 2 (equivalently, after adding a U to one of them, they can be interpreted as the Neron-Severi lattice and the transcendental lattice respectively for certain K3 surfaces, and thus one can view this as an instance of mirror symmetry for K3 surfaces, see [6]).

6.3 Deformations of Dolgachev Singularities and Periods of K3’s Looijenga [37, 38] has studied the deformation space of Dolgachev singularities. Briefly, they are unimodal, i.e. they have 1-parameter equisingular deformation. Within the equisingular deformation, there is a distinguished point corresponding to a singularity with C∗ -action (equivalently the equation is quasi-homogeneous). One can apply to that singularity Pinkham’s theory of deformations of singularities with C∗ -action. In this situation, there will be 1-dimensional positive weight direction (i.e. there is an induced C∗ action on the tangent space to the mini-versal deformation, and the weights refer to this action) corresponding to the equisingular deformations. The remaining (μ − 1) weights are negative and correspond to the smoothing directions. We denote by S− the germ corresponding to the negative weights. Because of the C∗ -action, S− can be globalized and identified to an affine space. Thus (S− \ {0})/C∗ is a weighted projective space of dimension μ − 2 (where μ is the Milnor number, e.g. μ − 2 = 10 for E12 ). The general theory of Pinkham states that (S− \ {0})/C∗ is to be interpreted as a moduli space of certain 2 dimensional pairs (X, H ) (H is to be interpreted as a hyperplane at infinity coming from a C∗ -equivariant compactification of the singularity). Looijenga [37, 38] observed that, in the case of Dolgachev singularities (with C∗ -action), the general point of (S− \ {0})/C∗ parametrizes a couple (X, H ) where X is a (smooth) K3 surface, and H is a Tp,q,r configuration of rational curves ((p, q, r) are the Dolgachev numbers of the singularity). In particular, the transcendental lattice of X is Tp ,q ,r ⊕ U (identified with the lattice of vanishing cycles for the triangle singularity), while Tp,q,r is its Neron–Severi lattice. In conclusion, the weighted projective space (S− \ {0})/C∗ is birational to a locally symmetric variety D/Γ corresponding to periods of Tp,q,r -marked K3 surfaces (the dimension is 20 − (p + q + r − 2) = 22 − (p + q + r) = p + q + r − 2 = μ − 2). Furthermore, Looijenga [38] showed that the structure of the Baily-Borel compactification (D/Γ )∗ is related to the adjacency of simple-elliptic and cusp singularities to the given Dolgachev singularity, and that the indeterminacy of the period map (S− \{0})/C∗  (D/Γ )∗ is related to the triangle singularities adjacent to the given one (e.g. E13 deforms to E12 and this will lead to indeterminacy, that is resolved by Looijenga’s theory; while, on the other hand E12 deforms only to simple elliptic, cusp, or ADE singularities, and thus there is no indeterminacy).

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Example 2 The simplest case is the deformation of E12 . The singularity has equation x 2 + y 3 + z7 = 0. In this situation, as explained, E12 only deforms to log canonical singularities giving a regular period map, which in turn gives an isomorphism: W P(3, 4, 6, 8, 9, 11, 12, 14, 15, 18, 21) ∼ = (S− \ {0}) /C∗ ∼ = (D/Γ )∗ . The weights above are the negative weights with respect to the C∗ -action on the tangent space to the mini-versal deformation of the singularity, which we recall can be identified with OC3 ,0 /J , where J := (x, y 2 , x 6 ) is the Jacobian ideal of f := x 2 + y 3 + z7 . In this example, (D/Γ )∗ is the Baily-Borel compactification for the moduli space of T2,3,7-marked K3 surfaces (N.B. T2,3,7 ∼ = E8 ⊕U ; also, because of self-duality in this case, the transcendental lattice is T2,3,7 ⊕ U = E8 ⊕ U 2 ).

6.4 Relating the Loci W6 , W7 and W8 to Z 6 , Z 7 and Z 9 Recall that W8 , W7 and W6 are the closures in M of the loci parametrizing polystable quartics with a singularity of type E12 , E13 and E14 respectively. The universal family of quartics gives a versal deformation for the E12 singularity (this follows from Urabe’s analysis [57] of quartics with this type of singularities, or more generally from du Plessis–Wall [7] and Shustin–Tyomkin [54]), thus at a quartic with E12 singularities such that the singularity has C∗ -action, the germ of (S− , 0) can be interpreted as the normal direction to W8 . Then, (S− \ {0}) /C∗ is nothing but the projectivized normal bundle, which is then the replacement via a (weighted) flip of the W8 locus. On the other hand, as noted in the example above, (S− \ {0}) /C∗ can be interpreted as the moduli of T2,3,7-marked K3s, which is the same as our Z 9 locus in F (the moduli of quartic K3 surfaces). The same considerations apply to the case of E13 and E14 singularities, but in those cases the identification of (S− \ {0}) /C∗ with the moduli of T2,4,5 (and T3,3,4 respectively) marked K3s (which then correspond to Z 7 and Z 6 respectively) involves one (or respectively two) flips (corresponding to the fact that E13 deforms to E12 , and similarly for E14 ). This is exactly as predicted in [34]. The argument above almost establishes our claim that a flip replace the Z 9 locus in F by W8 (the E12 locus) in M (and similarly for E13 and E14 ). In the following subsection, we strengthen the evidence towards this claim by a oneparameter computation (which shows that indeed the generic KSBA replacement for a quartic with E12 singularities (with C∗ -action) is a T2,3,7-marked K3). Example 3 Let [w, x, y, z] be homogeneous coordinates on a 3 dimensional projective space, and let X be the quartic defined by the equation x 2 w2 − 2xz2 w2 + y 3 w + x 3 z + z4 = 0.

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Computing partial derivatives, one finds that the singular set of X consists of the single point p := [1, 0, 0, 0]. In fact X has an E12 singularity at p, with C∗ action. To see why, we let w = 1, and hence (x, y, z) become affine coordinates. Then p is the origin, and a local equation of X near p is (x − z2 )2 + y 3 + x 3 z = 0. Let (s, y, z) be new analytic coordinates, where s = x − z2 ; the new equation is s 2 + y 3 + z7 + s 3 z + 3s 2 z3 + 3sz5 = 0. One recognizes s 2 + z7 + s 3 z + 3s 2 z3 + 3sz5 = 0 as an A6 singularity (assign weight 1/2 to s and weight 1/7 to z; since all other monomials appearing in the equation have weight strictly larger than 1, it follows that the equation is analytically equivalent to u2 + v 7 = 0), and hence in a neighborhood of p, the quartic X has analytic equation u2 + y 3 + v 7 = 0. This is exactly the local equation of an E12 singularity with C∗ action. Since X has no other singularity, it is stable by Shah, and [X] belongs to W8 .

6.5 The Semistable Replacement for Quartics with an E12 , E13 or E14 Quasi-Homogeneous Singularity We are assuming that we are given a quartic surface X0 with a unique Ek singularity (for k = 12, 13, 14) and such that the singularity has a C∗ -action (the singularity, in local analytic coordinates, is given by the equation in Table 3). We are considering a generic smoothing X /Δ and we are asking what is the KSBA replacement associated to this family. The computation is purely local, similar to that occurring in Hassett [17]. We will mimic the algorithm described in Example 1. A generic smoothing is locally given by V (f (x, y, z) + t) ⊂ (C4 , 0), where f is the local equation of the singularity as in Table 3. We make a base change t → t N so that the local monodromy is unipotent. Arnold et al. (see [4, Table on p. 113]) have computed the spectrum of the singularities for the simplest

Table 3 Equations of the relevant Dolgachev singularities Singularity E12 E13 E14

Equation (with C∗ -action) x 2 + y 3 + z7 = 0 x 2 + y 3 + yz5 = 0 x 3 + y 2 + yz4 = 0

Order N for base change 42 30 24

Weights (t, x, y, z) (1, 21, 14, 6) (1, 15, 10, 4) (1, 8, 12, 3)

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type of hypersurface singularities, including ours. The spectrum encodes the log of the eigenvalues of the local monodromy, thus from Arnold’s list it is immediate to find the base change giving unipotent monodromy; the relevant order N for the base change is given in Table 3 below. It turns out that the resulting threefold X = V (f (x, y, z) + t N ) ⊂ (C4 , 0) has a simple K3-singularity (analogue of simple elliptic) at the origin in the sense of Yonemura [59]. It follows that a suitable weighted blow-up of X at the origin will resolve this singularity, giving a K3 tail. The tail T will be one of the weighted K3 surfaces in the sense of M. Reid. What is specific in the situation analyzed here is that T has 3 singularities of type A lying on the exceptional divisor of the weighted blow-up (a rational curve). A routine analysis (see Gallardo [12] for further details) gives the following result. Proposition 15 Let X /Δ be a generic smoothing of a Dolgachev singularity of type Ek (k = 12, 13, 14). Then, after a base change of order N (as given in Table 3), followed by a weighted blow-up with weights as given in the table, gives a new 0 of X0 (with quotient central fiber X0 which is the union of the partial resolution X singularities given by the Dolgachev numbers (p, q, r)) and a K3 surface T with 3 singularities of types Ap−1 , Aq−1 , and Ar−1 liying on the (rational) curve C = 0 . Thus, the minimal resolution T of T is a Tp,q,r -marked K3 surface, where T ∩X (p, q, r) are the Dolgachev numbers of the Ek singularity. Proof The equation of the tail is simply V (f (x, y, z) + t N ) ⊂ W P(1, wx , wy , wz ) with f , N, and the weights as given in Table 3. Note that this is a weighted degree N hypersurface in a weighted projective space such that the sum of weights satisfies 1 + wx + wy + wz = N. This is precisely the K3 condition. Remark 21 Computations and arguments of similar nature have been done in the thesis of P. Gallardo (some of them appearing in [12]), who was advised by the first author. We have learned about similar computations done by Shepherd-Barron from an unpublished letter to R. Friedman. In conclusion we see that the replacement of the quartics with quasihomogeneous E12 , E13 , or E14 singularities are T2,3,7 , T2,4,5, T3,3,4 marked K3 surfaces respectively. These are parametrized by points of Z 9 , Z 8 , Z 7 respectively, see [34]. Specifically, we have the following result. Proposition 16 The loci Z 9 , Z 8 , Z 7 are naturally identified with the moduli spaces of T2,3,7 , T2,4,5, T3,3,4-polarized (in the sense of [6]) K3 surfaces respectively. Proof As already noted, for p1 + q1 + 1r < 1, Tp,q,r are hyperbolic lattices of signature (1, p + q + r − 3). Furthermore, the absolute value of their discriminant is pqr −   1 1 1 pq − pr − qr = pqr 1 − p − q − r (giving values 1, 2, 3 respectively in our

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situation). It follows, that the three Tp,q,r lattices considered here are isometric to E8 ⊕ U , E7 ⊕ U , and E6 ⊕ U respectively. Each of them has a unique embedding into the K3 lattice E82 ⊕ U 3 , and the corresponding orthogonal complements are E8 ⊕ U 2 , E8 ⊕ U 2 ⊕ A1 , and E8 ⊕ U 2 ⊕ A2 respectively. This coincides with our definition of the Z 9 , Z 8 , Z 7 loci from [34].

7 Looijenga’s Q-Factorialization The predictions of our previous paper [34] are concerned with the birational transformations that occur in the period domain F = D/Γ . Our working assumption is that all the modifications that occur at the boundary of the BailyBorel compactification F ∗ are explained by Looijenga’s Q-factorialization [41], together with the modifications occurring in F . More precisely we predict that, for 0 < 0 < 1/9, the birational map F (0 )  F ∗ is regular, small, and an isomorphism over F = D/Γ , and that the strict transform of Δ is a relatively ample of F ∗ by Q-Cartier divisor. In particular, we get a well defined birational model F setting F := F (0 ) for 0 < 0 < 1/9—this is Looijenga’s Q-factorialization. Our expectation is that, for a critical β ∈ [1/9, 1], the center of the birational map F (β − )  F (β + ) is the proper transform of the appropriate Z j appearing in (1) (Z 9 for β = 1/9, Z 8 for β = 1/7, and so on) via the birational map F (β − )  F . In particular, the above expectation predicts that the numbers of irreducible components of MI I and MI I I , and their dimensions, can be determined of Type II, Type III strata, once one has a description of the inverse images in F and their intersections with the strict transforms of the Z j ’s. In the present section we will spell out the predictions regarding MI I , and we will see that they match the computations of Sect. 4. Before we proceed with our computations, we note that there is a glaring discrepancy that seems to be against our predictions above: there are 8 Type II . In fact there is no contradiction, as components in M, while there are 9 in F we will see that the missing component is contained in the closure of one of the Z k ⊂ F strata and thus will disappear in the associated flip (and it will be hidden in the Type IV locus in M). We note that compared with the case of degree 2 K3 surfaces [39, 50] or cubic fourfolds [32, 42], this is a new phenomenon which points to the interesting nature of the quartic example.

7.1 Looijenga’s Q-Factorialization and Its Type II Boundary Components The locally symmetric variety F has at worst finite quotient singularity, and thus it is Q-factorial. Since the boundary F ∗ \ F is of high codimension, any divisor of F

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extends uniquely as a Weil divisor, but typically not a Q-Cartier divisor. Looijenga [41] has constructed a Q-factorialization associated to any arithmetic hyperplane arrangement (or equivalently pre-Heegner divisor in the terminology of [34]). Here, we are interested in the Q-factorialization of the closure of Δ = 12 (Hu + Hh ). → F ∗ be the Looijenga Q-factorialization associated to the Definition 6 Let F hyperplane arrangement H = π −1 (Hh ∪ Hu ) (where π : D → D/Γ is the natural projection) of hyperelliptic and unigonal pre-Heegner divisors. From our perspective, it is immediate to see that the Q-factorialization coincides with one of our models: → Proposition 17 Let 0 <   1. Then the composition of birational maps F ∼ ∗ ∗ F and F  F () is an isomorphism F −→ F (). has the property that λ + Δ extends to a Q-Cartier and Proof By construction, F (N.B. the relative ampleness of Δ is not explicitly stated ample divisor class λ +  Δ. in [41], but this is precisely what Looijenga checks). Hence the ring of sections , → F ∗ is a small makes sense (and is finitely generated). Since F R(F λ +  Δ) map, and F is normal (by construction), the restriction of sections to F ⊂ F ∼ defines an isomorphism R(F , λ +  Δ) = R(F , λ + Δ). Remark 22 According to the discussion of [28, Ch. 6], the Q-factorialization of Δ is unique: it is either F () or F (−) (depending on the requested relative ampleness). The main issue is that the finite generation of the ring of sections defining F () is not a priori guaranteed. Looijenga [41] makes use of the special structure of the Baily-Borel compactification (e.g. the tube domain structure near the boundary, and the existence of toroidal compactifications) to obtain that the Q-factorialization is well defined, and furthermore to get an explicit description of it. Remark 23 The results of [34] (see esp. Proposition 5.4.5) predict that the above proposition holds for 0 <  < 19 . The stratification F ∗ = F I  F I I  F I I I defines by pull-back a stratification =F I  F I I  F I I I . The boundary strata of F are the irreducible components F of the above strata. We are interested in the number of the Type II boundary strata and their dimensions. of F We start by recalling that the structure of the Baily-Borel compactification for quartic surfaces was worked out by Scattone [48]: there are 9 Type II boundary components, and a single Type III boundary component. Proposition 18 (Scattone [48]) The boundary of the Baily-Borel compactification F ∗ of the moduli space of quartic surfaces consists of 9 Type II boundary components, and a single Type III component. The Type II boundary components are naturally labeled by a rank 17 negative definite lattice as follows: D17 , D9 ⊕E8 , D12 ⊕D5 , D3 ⊕(E7 )2 , A15 ⊕D2 , , A11 ⊕E6 , (D8 )2 ⊕D1 , D16 ⊕D1 , and (E8 )2 ⊕D1 respectively.

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Table 4 Dimension of the boundary strata in F

D17 D3 ⊕ (E7 )2 (D8 )2 ⊕ D1

1 4 2

D9 ⊕ E8 A15 ⊕ D2 D16 ⊕ D1

10 3 6

D12 ⊕ D5 A11 ⊕ E6 (E8 )2 ⊕ D1

6 1 2

are Proposition 19 The dimensions of the Type II strata in the compactification F given in Table 4. Proof A type II boundary component is determined by the choice of an isotropic rank 2 primitive sublattice E ⊂ Λ(∼ = E82 ⊕ U ⊕ *−4,) (up to the action of the monodromy group). The label associated to a Type II boundary component is the ⊥ /E (with the root sublattice contained in the negative definite rank 17 lattice EΛ convention of including also D1 = *−4, in the root lattice). According to [48], this is a complete invariant for a Type II boundary component in the case of quartic surfaces. The construction of Looijenga [41] (see esp. Section 3 and Proposition 3.3 of loc. cit.) depends on the linear space   L := ∩H ∈H ,E⊂H (H ∩ E ⊥ ) /E ⊂ E ⊥ /E. ⊥ /E. Note M is a negative definite rank 17 lattice. Then, More precisely, let M := EΛ we recall that the fiber over a point j in the type II boundary component (recall each Type II boundary component is a modular curve, here h/SL(2, Z)) associated to E is simply the quotient of the abelian variety J (Ej ) ⊗Z M by a finite group (here Ej denotes the elliptic curve of modulus j , and J (Ej ) its Jacobian). What Looijenga has observed is that L is the null-space of the restriction to the toroidal boundary (of Type II) of the linear system determined by the hyperplane arrangement H . And thus, the fiber for the Q-factorialization (which as discussed above corresponds to the Proj of the ring of sections of λ + Δ; also recall (the pull-back of) λ restricts to trivial on the toroidal boundary) over the point j in the Type II boundary component associated to E is (up to finite quotient) J (Ej ) ⊗Z M/L. Now, we recall that the lattice Λ can be primitively embedded into the Borcherds lattice I I2,26 with orthogonal complement D7 (in a unique way). We fix

Λ → I I2,26 and R = Λ⊥ ∼ = D7 . With respect to this embedding, a hyperelliptic hyperplane corresponds to an extension of R to a (primitively embedded) D8 into I I2,26 , while a unigonal divisor to a E8 . Successive intersections of hyperplanes from H correspond to extensions of R = (D7 ) into Dk lattices. Similarly, if E is rank 2 isotropic (primitively embedded), then we recall that M = E ⊥ /E can be embedded into one the 24 Niemeier lattices (i.e. rank 24 negative definite even unimodular lattices) with orthogonal complement D7 . The same considerations as before apply: a hyperelliptic divisor correspond to an extension to D8 (and

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repeated intersections to D7+k ), while a unigonal one corresponds to an extension to E8 . By inspecting the possible embeddings of Dk lattices into Niemeier lattices, one obtains the dimensions claimed in Table 4. The only exception is the case D17 (in which case D7 extends to D24 ) for which L = 0 ⊂ M, and thus the Heegner divisor is already Q-Cartier (and no modification is necessary; see [41, Cor. 3.5]).

7.2 Matching Type II Strata In order to understand the matching of the GIT and Baily-Borel Type II strata, one needs to consider a generic smoothing X /Δ of a Type II quartic surface X0 and compute the limit MHS with Z-coefficients. The analogous case of K3 surfaces of degree 2 was analyzed by Friedman in [10]. Inspired by Friedman’s analysis, we make the following definition: Definition 7 Let X0 be a Type II polystable quartic surface. The associated (isomorphism class of) lattice is the direct sum of the following lattices: 1. 2. 3. 4.

r singularity of X. One copy of Er for each E One copy of D4d+4 for each degree d rational curve in the singular set of X. One copy of A4d−1 for each degree d elliptic curve in the singular set of X. ˜ where X˜ is the minimal resolution of the The lattice *hX˜ , KX˜ ,⊥ ⊂ Pic(X) normalization of X and hX˜ is the polarization class on X˜ (e.g. if X˜ is a degree 2 del Pezzo with the anticanonical polarization, we add E7 ).

Remark 24 To understand the meaning of the lattice associated to a Type II degeneration X0 , one needs to consider a generic smoothing X /Δ of X0 , followed /Δ. The lattice introduced in the by a semi-stable (Kulikov type) resolution X definition above is essentially (W2 /W1 )prim from [10, (5.1)]. The main point here (similar to the discussion of Sect. 6) is that one has quite a good understanding of the semistable replacement in the Type II case. For instance, the simple elliptic r (r = 6, 7, 8) will be replaced by degree 9 − r del Pezzo tails (this singularities E leads to the first item of Definition 7). Remark 25 By going through our list of Type II components of M, one checks the following: 1. Two polystable quartic surfaces belonging to the same Type II component of M have isomorphic associated lattices. 2. The lattice associated to a polystable quartic surface of Type II has rank 17, is negative definite, even, and belongs to the list of lattices associated to Type II boundary components of F ∗ , see Proposition 18. 3. By associating to a Type II component of M the lattice associated to any polystable quartic in the component (see Item (1)), we get a one to one

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correspondence between the set of Type II components of M and the set of lattices appearing in Proposition 18, provided we remove the D17 lattice. The geometric meaning of the lattice associated to a polystable quartic of Type II is provided by our next result, which is proved by mimicking the arguments of Friedman in [10] (see esp. [10, Rem. 5.6]). Proposition 20 Let X be a polystable Type II quartic surface. The period point p([X]) belongs to the Baily-Borel Type II boundary component labeled by the lattice associated to X. So far we have proved that the set of lattices appearing in Proposition 18, once we remove the D17 lattice, parametrizes both the components of MI I , and the Type II boundary components of F ∗ , with the exclusion of one. The two parameterizations are compatible with respect to the period map. Of course the same set of lattices , with the exception of one. parametrizes Type II boundary components of F Proposition 21 Let L be one of the lattices appearing in Proposition 18, with the indexed exception of D17 . The dimension of the Type II boundary component of F by L is equal to the dimension of the Type II component of M indexed by the same L. Proof We illustrate the computation of dimensions in the highest dimensional case: 8 singularity such that no line passes through II(5), i.e. quartics that have a single E this singularity. Let X0 be a generic surface of this type; then X0 has a singularity of 8 at some point p and is smooth away from p, see Remark 9. Let X 0 → X0 type E be the minimal resolution. By Remark 9, the exceptional divisor D is an elliptic 0 ) = 11 curve with self-intersection −1, D is an anti-canonical section, and ρ(X 0 is the blow-up of P2 along 10 points on an elliptic curve). Thus, H 2 (X 0 ) is (i.e. X nothing else than the lattice I1,10 . By the discussion in Remark 10, it follows that ⊥  *KX 0 + hX 0 ,H 2 (X 0 ) is isometric to D9 . Since X0 has an E8 singularity, by our rule (Definition 7), the associated label is D9 ⊕ E8 . From Shah [51], a generic surface X0 of Type II(5) is GIT stable. On the other hand, the results of [7] and [54] imply in particular (loc. cit. give general conditions in terms of total Tjurina number) that the universal family of quartic 8 singularity. From these two results, it follows that surfaces versally unfolds the E 8 singularity is 10 = μ(E 8 ) (where μ is the codimension of the locus with a fixed E the Milnor, and also, in this case, the Tjurina number), but there is an additional 1dimensional deformation corresponding to moduli of simple elliptic singularities. Summing up, the II(5) locus has codimension 9 (i.e. dimension 10) in the GIT quotient M. (The same dimension count also follows from the geometric description given in Remark 9.) has The computation of the dimension of the stratum labeled by E8 ⊕ D9 in F been carried out in Proposition 19. Here we point out that this case corresponds to the Niemeier lattice containing the root system D16 ⊕ E8 . In that situation, the maximally embedded Dl is D16 , which means (using the notation of Proposition 19) → F ∗ over a point j in the dim M/L = 9 (N.B. 16 = 7 + 9). Thus the fiber of F Type II component labeled by E8 ⊕ D9 is 9. Then again, by varying j , we obtain

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; thus the dimensions in M and F a 10-dimensional component (this time in F match). To get further geometric understanding of the matching of the GIT com , we note that ponent II(5) and of the component labeled by D9 ⊕ E8 in F there exists an extended period map. Specifically, recall that X0 carries a mixed Hodge structure (MHS), and that there exists also a limit mixed Hodge structure (LMHS). The Baily-Borel compactification F ∗ encodes the graded pieces of the LHMS (in this situation, the modulus of the elliptic curve C, and a discrete part, i.e. the choice of Type II component, or equivalently the label of the component). As previously discussed, the graded pieces of the LMHS can be read off from those of the MHS on degeneration X0 (the weight 1 part follows from Theorem 2, while the discrete weight 2 part is the rule given Σ by Definition 7). On the other hand, a toroidal compactification F (which is unique over the Type II stratum) encodes the full LMHS (i.e. the graded pieces, plus the extension data; see Friedman [10] for a full discussion). Finally, the semitoric compactifications of Looijenga are sitting between the Baily-Borel and Σ → F ∗ . Thus, from a Hodge theoretic the toroidal compactifications: F → F perspective, F retains the graded pieces of the LHMS, plus partial extension data. As explained below, this partial extension data is exactly the extension data that can be read off from the central fiber X0 (without passing to the Kulikov model). Specifically, in the case that we discuss here (Type II(5)), the Kulikov model 0 ∪E T , where (as above) X 0 is the resolution of the quartic surface with is X  an E8 singularity, T is a “tail” (depending on the direction of the smoothing). In this situation, T is a degree 1 del Pezzo surface, whose primitive cohomology is 0 is a rational surface with primitive cohomology D9 . Finally, the gluing E8 . X 0 ), which curve E is an elliptic curve (with self-intersection 1 on T and −1 on X gives the modulus j discussed above. Fixing j , the modulus of these type of surfaces (up to the monodromy action) is the 17 dimensional abelian variety (E8 ⊕ D9 ) ⊗Z J (Ej ) (this is precisely the fiber of the toroidal compactificaΣ

→ F ∗ over the appropriate Type II Baily-Borel boundary point). tion F → F ∗ becomes When passing to Looijenga Q-factorialization, the fiber of F D9 ⊗Z M/L (N.B. M/L = D9 in this case). This fiber can be identified with 0 , D) with fixed j -invariant for D). the moduli space of X0 (or equivalently (X More precisely, it is possible to see that the restriction of the extended period map → F∗ M  F (which extends over the Type II and III locus) to the locus II(5) is nothing 0 , D) (see [13] and [11] else but the period map for the anticanonical pair (X for a general modern discussion of the period map for anticanonical pairs, and [57, Section 5] for the specific case discussed here; all of this originates with work of Looijenga [36]). In conclusion, we get a perfect matching between the

280 Table 5 Matching of the Type II strata

R. Laza and K. G. O’Grady GIT stratum II(1) II(2) II(3) II(4) II(5) II(6) II(7) II(8)

BB stratum (E8 )2 ⊕ D1 (E7 )2 ⊕ A3 (D8 )2 ⊕ D1 E6 ⊕ A11 E8 ⊕ D9 D12 ⊕ D5 D16 ⊕ D1 A15 ⊕ (A1 )2

Dimension 2 4 2 1 10 6 2 3

labeled by D9 + E8 Type II(5) stratum in M and the Type II stratum in F 2 (Table 5). Remark 26 (Kulikov Models) It is not hard to produce Kulikov models for each of the Type II degenerations above. For instance, in a semi-stable degeneration, each r singularities will be replaced by a del Pezzo of degree (9 − r). As an of the E 8 singularities will give example, the case II(1) corresponding to a quartic with 2 E 2 degree 1 del Pezzo surfaces, glued to an elliptic ruled surface (which is in the fact the resolution of the singular quartic; the del Pezzo surfaces are “tails” induced by the smoothing; see Sect. 6.5 above for related computations). The case of quartics singular along a curve typically are obtained by projection from a rational surface (frequently del Pezzo). For instance the case II(6) is obtained by projecting a degree 4 del Pezzo from a point in P4 . The associated label D12 ⊕ D5 has the following meaning: D5 (= E5 ) is the primitive cohomology of the associated degree 4 del Pezzo. On the other hand D12 is coming from the singular locus of the quartic (in this case a conic) and the rule (Definition 7) given above.

7.3 The Missing GIT Type II Component The reader might be puzzled by the fact the BB stratum corresponding to D17 does not occur in the list of Type II components of M. We can explain this as follows. First of all as noted in Proposition 19, along this stratum the hyperelliptic divisor is Q-factorial and thus it is not affected by the Q-factorialization. Moreover, this is precisely the boundary component that is contained in all elements of the Dtower. (this is the component that survives when we go to low dimensions). As discussed the entire Δ(k) (when we get to codimension 9) is contracted and then flipped (i.e. there is no deeper flip). This is indeed compatible with a theorem of

2 In fact, while we do not check it here, we expect that the extended period map is an isomorphism (at the generic points) between the II(5) and II(D9 + E8 ) strata (and similarly for the other Type II strata).

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Looijenga which identifies F (10)∗ with a certain weighted projective space and with the moduli space of T2,3,7 (= E8 ⊕ U ) marked K3s (see Sect. 6). In conclusion, the 9th boundary component is flipped all at once together with a big stratum, and thus will not be visible in MI I . It is “hidden” in the E12 stratum (i.e. IV(8)) in MI V . Acknowledgements While the same acknowledgements as in [34] apply here, we repeat our thanks to E. Looijenga for his inspiring work and some helpful feedback, and to IAS for hosting us during the Fall of 2014, which represents the start of our investigations. Finally, much of this paper was written while the first author was on sabbatical and visiting École Normale Supérieur. He thanks ENS for hospitality and Simons Foundation and Fondation Sciences Mathématiques de Paris for partially supporting this stay. Research of the first author is supported in part by NSF grants DMS-125481 and DMS1361143. Research of the second author is supported in part by PRIN 2013.

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Generalized Boundary Strata Classes Aaron Pixton

Abstract We describe a generalization of the usual boundary strata classes in the Chow ring of M g,n . The generalized boundary strata classes additively span a subring of the tautological ring. We describe a multiplication law satisfied by these classes and check that every double ramification cycle lies in this subring.

1 Introduction Let g, n ≥ 0 satisfy 2g − 2 + n > 0. These notes will be concerned with certain classes in the Chow ring of the moduli space M g,n of stable curves of genus g with n marked points. All of our classes will be tautological; that is, they are contained in a certain subring R ∗ (M g,n ) ⊆ A∗ (M g,n ), the tautological ring. The full structure of the tautological ring is not well understood, though a large family of relations is known [7]. The double ramification (DR) cycle is an important class in R ∗ (M g,n ), defined in terms of the virtual class in relative Gromov-Witten theory. Informally, the DR cycle parametrizes curves admitting a map to P1 with specified ramification profiles over two points. Faber and Pandharipande [4] proved that the DR cycle lies in the tautological ring, and then an explicit formula for the DR cycle in terms of basic tautological classes was recently proven in [6]. Trying to simplify this formula was one of the main motivations for these notes. We will define the boundary subring B ∗ (M g,n ) of the tautological ring ∗ R (M g,n ). The boundary subring is defined as the additive span of certain generalized boundary strata classes [Γ ] corresponding to topological types of prestable curves. The boundary subring has a convenient multiplication law and

A. Pixton () Massachusetts Institute of Technology, Cambridge, MA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2018 J. A. Christophersen, K. Ranestad (eds.), Geometry of Moduli, Abel Symposia 14, https://doi.org/10.1007/978-3-319-94881-2_9

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may be of interest in its own right. The DR cycle is contained in the boundary subring and admits a simpler formula in terms of these generalized boundary strata classes. In Sect. 2, we recall some of the basic definitions and properties of the tautological ring and fix some notation for dual graphs. In Sect. 3, we define the generalized boundary strata classes and describe the multiplication law in the boundary subring. In Sect. 4, we define the genus-free boundary subring, a natural subring of the boundary subring. Finally, in Sect. 5 we explain how to write the double ramification cycle in terms of generalized boundary strata classes.

2 The Tautological Ring Following Faber and Pandharipande [3], the tautological rings R ∗ (M g,n ) are defined simultaneously for all g, n ≥ 0 satisfying 2g − 2 + n > 0 as the smallest subrings of the Chow ring A∗ (M g,n ) closed under pushforward by forgetful maps M g,n+1 → M g,n and gluing maps M g,n+2 → M g+1,n or M g1 ,n1 +1 × M g2 ,n2 +1 → M g1 +g2 ,n1 +n2 . In this section we will recall a more explicit description of the tautological ring given by Graber and Pandharipande [5, Appendix A]. They described a set of additive generators and a multiplication law satisfied by these generators. The generators can be indexed by connected finite graphs decorated with certain additional structures. In this paper we will be interested in several slightly different types of decorated graphs, so we go through these structures one at a time. We begin by letting a graph with n legs mean a connected finite graph Γ along with a sequence of vertices (possibly with repetition) l1 , l2 , . . . , ln ∈ V (Γ ) of length n ≥ 0. Here li is the location of the ith leg of Γ . We think of legs as half-edges, so the valence nv of a vertex v includes a contribution of one for each li equal to v. Then a graph of genus g with n legs is a graph Γ with n legs along with a nonnegative integer gv for each vertex v ∈ V (Γ ), satisfying the equality g = h1 (Γ ) +



gv ,

v∈V (Γ )

where h1 (Γ ) = |E(Γ )| − |V (Γ )| + 1 is the cycle number of Γ . Next, a graph Γ of genus g with n legs is stable if it satisfies the condition 2gv − 2 + nv > 0 at each vertex v ∈ V (Γ ). We will usually just call such a graph a stable graph and treat the values of g and n as understood.

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Stable graphs correspond to strata in the boundary stratification of M g,n as follows. First, given a stable graph Γ we write M Γ :=



M gv ,nv .

v∈V (Γ )

The edges of Γ then define a gluing map ξΓ : M Γ → M g,n . The image of this map is the closed boundary stratum corresponding to Γ , the closure of the locus of curves with dual graph Γ . Up to a factor of | Aut(Γ )|, the class of this stratum is equal to the pushforward ξΓ ∗ 1. The Chow ring of each factor M gv ,nv of M Γ contains basic tautological classes: the Arbarello-Cornalba [1] kappa classes κj (j > 0) of codimension j and the cotangent line classes ψi (1 ≤ i ≤ nv ) of codimension 1. We can then consider classes ξΓ ∗ α ∈ A∗ (M g,n ) given any stable graph Γ and any monomial α in the kappa and psi classes on the factors of M Γ . These are precisely the additive generators for the tautological ring considered in [5]. In other words, the generators can be indexed by the data of a stable graph Γ with additional kappa and psi decorations (on the vertices and half-edges respectively). We will describe this as a stable graph Γ with a monomial decoration α. The multiplication rule for these classes given by Graber and Pandharipande [5, eq. (11)] has a somewhat complicated form. Let Γ1 , Γ2 be two stable graphs with monomial decorations α1 , α2 respectively. Then (ξΓ1 ∗ α1 ) · (ξΓ1 ∗ α2 ) =

 Γ, α1 , α2 ,E1 ,E2

1 c · ξΓ ∗ ( α1 α2 Ψ ). Aut(Γ )

(1)

This formula requires some explanation. First, the sum runs over (representatives of isomorphism classes of) stable graphs Γ along with two monomial decorations  α1 , α2 on Γ and two disjoint sets of edges E1 , E2 ⊆ E(Γ ) such that αi does not use any psi classes located on the edges in Ei . The coefficient c ∈ Z≥0 then counts the number of ways to choose isomorphisms (of monomial-decorated graphs) (Γ /Ei , αi ) ∼ = (Γi , αi ) for i = 1, 2, where Γ /Ei is the stable graph formed by contracting the edges in Ei (and combining vertex genera in the natural way, including adding one whenever a loop is contracted). The factor Ψ is the product of (−ψ1 − ψ2 ) over all edges in E(Γ ) \ (E1 ∪ E2 ), where ψ1 , ψ2 are the ψ classes on the two sides of the edge.

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Note that the multiplication law (1) is quite complicated even when the monomials α1 , α2 are both equal to 1. In particular, the product of two boundary strata classes (ξΓ1 ∗ 1) · (ξΓ2 ∗ 1) is not necessarily a sum of boundary strata classes because of the Ψ factor. The generalized boundary strata classes defined in the next section will simultaneously fix this issue and yield a simpler multiplication law.

3 Generalized Boundary Strata Classes The goal of this section is to define classes [Γ ] ∈ R d (M g,n ) for any graph Γ of genus g with n legs (with 2g − 2 + n > 0), where d = |E(Γ )| is the number of edges in Γ . The definition will satisfy [Γ ] = ξΓ ∗ 1 for any stable graph Γ . When Γ is unstable, [Γ ] will provide a convenient generalization of the usual boundary strata classes. For example, if Γ is the graph with two vertices connected by a single edge, and one of the vertices is genus 0 and has leg i and no other legs, then we will have [Γ ] = −ψi , the first Chern class of the line bundle on M g,n given by the tangent space at the ith marked point. We define [Γ ] in two steps. First we define [Γ ] for graphs of genus g with n legs satisfying the semistability condition 2gv − 2 + nv ≥ 0 at every vertex v ∈ V (Γ ). In other words, the only type of unstable vertex that can occur is a genus 0 vertex of valence 2. These unstable vertices are naturally partitioned into paths, and there are two different types of paths of unstable vertices: a path of k ≥ 1 unstable vertices between two stable vertices v1 , v2 (possibly equal) or a path of k ≥ 1 unstable vertices between a stable vertex v and a leg labeled i. In the first case, we contract the path to a single edge between v1 and v2 and decorate it with (−ψ1 − ψ2 )k /(k + 1)!. In the second case, we contract the path to a single leg on v labeled i and decorate it with (−ψ)k /k!. After performing these actions to every path of unstable vertices, we are left with a stable graph Γcont decorated by a polynomial α in the psi classes on M Γcont . We then set [Γ ] := ξΓcont ∗ α, so [Γ ] is a linear combination of the usual tautological ring generators described in Sect. 2. Now we define [Γ ] for general graphs Γ of genus g with n legs. Let v1 , . . . , vm be the vertices of Γ with genus 0 and valence 1 (in some order). Create a new graph Γ by attaching new legs labeled n + 1, . . . , n + m to v1 , . . . , vm respectively (i.e. ln+i = vi ). Then 2gv − 2 + nv ≥ 0 at every vertex v ∈ V (Γ ), so we can use our previous definition to obtain a class [Γ ] ∈ R d (M g,n+m ). Then take [Γ ] := (πm )∗ ((−ψn+1 ) · · · (−ψn+m )[Γ ]) ∈ R d (M g,n ),

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where πm : M g,n+m → M g,n is the map forgetting the last m markings. (This pushforward will create kappa classes in general: see [1, eq. (1.12)]). This completes the definition of [Γ ] for any graph Γ of genus g with n legs. We now let B ∗ (M g,n ) ⊆ R ∗ (M g,n ) be the additive span of the classes [Γ ] inside the tautological ring. We have the following multiplication law, a much simpler version of (1): Proposition 1 Let Γ1 , Γ2 be two graphs of the same genus g and number of legs n. Then [Γ1 ] · [Γ2 ] =

 Γ

1 cΓ ,Γ ,Γ [Γ ], | Aut(Γ )| 1 2

where cΓ1 ,Γ2 ,Γ is the number of ways of choosing a partition of the edges E(Γ ) = E1  E2 along with isomorphisms Γ /Ei ∼ = Γi for i = 1, 2. Proof This follows from (1) along with the definition of [Γ ]. We sketch part of the proof here to give the idea. Suppose that we have chosen a graph Γ along with a choice of the data counted by cΓ1 ,Γ2 ,Γ (i.e. E1 , E2 , and isomorphisms Γ /Ei ∼ = Γi ). We want to match the contribution [Γ ] with (part of) the general multiplication law (1). First, if Γ is a stable graph then Γ1 , Γ2 are also stable (since contractions of a stable graph are stable). Then [Γi ] = ξΓi ∗ 1 and the same data (E1 , E2 , isomorphisms) describes a term [Γ ] in (1). Next, suppose Γ is a semistable graph. Then as previously explained, the unstable vertices can be partitioned into paths of two types: those between two stable vertices and those between a stable vertex and a leg. In either case, suppose such a path consists of m edges. Then these edges are partitioned between E1 and E2 ; suppose that mi of them are in Ei , so m1 + m2 = m. For the contracted graphs Γ /Ei the values   of m1 and m2 are all that matter, so the contribution appears with multiplicity mm1 . Then the matching with (1) follows from the simple identities % & m (−ψ1 − ψ2 )m1 −1 (−ψ1 − ψ2 )m2 −1 (−ψ1 − ψ2 )m−1 · · (−ψ1 − ψ2 ) = · m1 m1 ! m2 ! m! and % & m (−ψ)m1 (−ψ)m2 (−ψ)m · = · m1 m1 ! m2 ! m! for the two types of paths. In the first case, the extra factor of (−ψ1 − ψ2 ) comes from the factor Ψ in (1).

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The case where Γ is non-semistable is handled similarly. Here the definition of [Γ ] produces polynomials in the kappa classes (via the pushforward formula for psi class monomials, see [1, eq. (1.13)]), and a slightly more complicated combinatorial identity is needed to get the desired matching with terms in (1); we omit this for brevity.  ) We conclude that B ∗ (M g,n ) is a ring. We call it the boundary subring. Let Sg,n be a Q-vector space with basis indexed by isomorphism classes of graphs of genus g with n legs. The multiplication law above (Proposition 1) defines a multiplication operation on Gg,n . It is easily checked that this formal graph algebra Gg,n is a commutative, associative graded Q-algebra with unit given by the graph with no edges, and then [−] : Gg,n → R ∗ (M g,n ) is a homomorphism with image B ∗ (M g,n ). Remark It is worth noting that the multiplication law becomes even simpler if we rescale our generalized boundary strata class by a factor of | Aut(Γ )|, since we can rewrite it as  [Γ1 ] [Γ2 ] [Γ ] · = , cΓ 1 ,Γ2 ,Γ | Aut(Γ1 )| | Aut(Γ2 )| | Aut(Γ )| Γ

where cΓ 1 ,Γ2 ,Γ is the number of ways of choosing a partition of the edges E(Γ ) = E1  E2 such that Γ /Ei is isomorphic to Γi for i = 1, 2. This also makes it easier to check that the multiplication is associative. However, our chosen normalization makes the statement of Proposition 2 below much nicer. The most important property of the classes [Γ ] has already been discussed: they generate a subring B ∗ (M g,n ) with a relatively simple multiplication law. They also satisfy simple identities under pullback and pushforward by forgetful maps: Proposition 2 Let g, n ≥ 0 satisfy 2g − 2 + n > 0. Let Γ be a graph of genus g with n legs. Then we have: 1. If π : M g,n+1 → M g,n is the map forgetting marking n + 1, then π ∗ [Γ ] =



[Γv ],

v∈V (Γ )

where Γv is formed from Γ by attaching leg n + 1 at vertex v. 2. If n > 0 and π : M g,n → M g,n−1 is the map forgetting marking n, then π∗ (ψn [Γ ]) = (2gv − 2 + nv )[Γ ], where Γ is the graph formed from Γ by removing the nth leg and gv , nv are the genus and valence in Γ at the vertex where the leg was removed.

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As with Proposition 1, this proposition is a straightforward combinatorial consequence of the analogous, more complicated results for general tautological classes.

4 Genus-Free Boundary Classes We will now define a natural subring of the boundary subring B ∗ (M g,n ). The idea is to sum over all possible distributions of the available genus across the vertices of the graph. Let G be a graph with n legs (but no genus assignments). For any nonnegative integer g satisfying 2g − 2 + n > 0, we define 

[G]g :=

[Γ ] ∈ R ∗ (M g,n ),

g# v ≥0 for v∈V (G) gv =g−h1 (G)

where the Γ inside the sum is the genus-labeled graph given by G and the gv . We then immediately have from Proposition 1 that these genus-free boundary classes additively span a subring GB ∗ (M g,n ) ⊆ B ∗ (M g,n ), with the following multiplication law: Proposition 3 Let n ≥ 0 and let G1 , G2 be two graphs with n legs. Then for any g ≥ 0 satisfying 2g − 2 + n > 0, we have [G1 ]g · [G2 ]g =

 G

1 cG ,G ,G [G]g , | Aut(G)| 1 2

where cG1 ,G2 ,G is the number of ways of choosing a partition of the edges E(G) = E1  E2 along with isomorphisms G/Ei ∼ = Gi for i = 1, 2. We call this subring GB ∗ (M g,n ) the genus-free boundary subring. As before, we can define a formal graph algebra Gn (now without genus information in the graphs) and view [−]g as a homomorphism from this algebra to GB ∗ (M g,n ). Since the structure of Gn is genus-invariant, it is natural to make the following stability conjecture about the genus-free boundary subring: Conjecture 1 For fixed d, n ≥ 0, the homomorphism [−]g : Gn → GB ∗ (M g,n ) is an isomorphism in degree d for all sufficiently large g. For example, for d = n = 1 we do not have an isomorphism for g = 1 since then [Γ ]1 = κ1 − ψ1 = 0 ∈ R ∗ (M 1,1 ) for Γ the unique connected graph with two vertices, one edge, and one leg. But then for g ≥ 2 it is easily checked that the map [−]g is an isomorphism in degree 1. Remark We’ve now defined three subrings of the Chow ring: GB ∗ (M g,n ) ⊆ B ∗ (M g,n ) ⊆ R ∗ (M g,n ).

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It is reasonable to ask whether they are actually distinct. The difference between B ∗ (M g,n ) and R ∗ (M g,n ) is that psi classes on opposite sides of an edge must appear together in the boundary subring, so in small genus (e.g. g ≤ 3) we can use tautological relations to remove most psi classes and see that B ∗ (M g,n ) = R ∗ (M g,n ). For larger values of g, say g ≥ 6, these rings should become different. In contrast, already for M 1,3 the subring GB ∗ (M g,n ) is strictly smaller than the other two.

5 Rewriting the Double Ramification Cycle Formula Let a1 , . . . , an be integers with sum zero. Taking the positive ai and the negative ai separately gives two partitions μ, ν of the same number k. Also, let n0 count the number of ai equal to zero. The double ramification cycle DRg (a1 , . . . , an ) ∈ R g (M g,n ) can then be viewed as parametrizing curves admitting a degree k map to P1 with ramification profiles μ, ν over points 0, ∞ respectively and n0 internal marked points. More precisely, let M g,n0 (P1 /{0, ∞}, μ, ν)∼ be the moduli space of stable relative maps to a rubber P1 (see [4, Section 0.2.3]). Then DRg (a1 , . . . , an ) = p∗ [M g,n0 (P1 /{0, ∞}, μ, ν)∼ ]vir , the pushforward of the virtual class along the natural map p : M g,n0 (P1 /{0, ∞}, μ, ν)∼ → M g,n . A complicated but explicit formula for DRg (a1 , . . . , an ) in terms of tautological classes was recently proven in [6, Theorem 1]. In this section we explain how this formula can be simplified by means of the genus-free boundary classes [G]g from Sect. 4. The DR cycle formula was written in [6, Section 0.4.2] as a sum over stable graphs decorated by certain power series in the psi classes. The psi classes appear in these power series either as the psi class associated to a leg of the graph or as the sum of the two psi classes associated to an edge of the graph. These are precisely the two types of psi decorations that appear in generalized boundary strata, and it turns out that we can simply remove the psi power series and enlarge the sum to run over the unstable graphs as well as the stable graphs. In addition, the coefficients in this sum do not depend on how the genus is distributed over the vertices, so we can pass to the genus-free boundary strata classes. The result is the following formula. Let d ≥ 0 and r > 0 be integers. Then we can define DRd,r g (a1 , . . . , an ) = ⎛   ⎝ 1 h (G) 1 r G



w:H (G)→{0,...,r−1} e=(h,h )∈E(G)

⎞ [G]g 1 w(h)w(h )⎠ , 2 | Aut(G)|

(2)

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where the first sum is over all graphs with n legs and exactly d edges and the second sum is over all weightings mod r of G, as defined in [6, Section 0.4.1]. Then the actual DR cycle DRg (a1 , . . . , an ) is the constant term in r of g,r DRg (a1 , . . . , an ) (which is polynomial in r for r sufficiently large as explained in [6]). In particular, DRg (a1 , . . . , an ) belongs to the genus-free boundary subring GB ∗ (M g,n ). Remark There are two natural generalizations of the DR cycle formula that are discussed in [6]. We will briefly discuss how they fit into our boundary ring framework. First, the DR cycle formula is really the degree g part of a class of impure degree. The parts of this formula in degrees greater than g give tautological relations, as conjectured in [6] and proved by Clader and Janda [2]. These relations in degree d > g are then just the constant term in r of the class DRd,r g (a1 , . . . , an ) in (2). In other words, the r-constant term of the formula (2) can be interpreted as defining a class DRd (a1 , . . . , an ) in the formal graph algebra Gn from Sect. 4, and this class has the property that its image in GB ∗ (M g,n ) is equal to the DR cycle for g = d and equal to zero for g < d. Second, there is a k-twisted version of the DR cycle formula [6, Section 1.1]. This formula yields a class in the boundary subring B ∗ (M g,n ) but no longer in the genus-free boundary subring GB ∗ (M g,n ). Formula (2) will still hold with two modifications: G must be replaced by a graph Γ of genus g with n legs and w must now be a k-weighting mod r (this condition for k = 0 depends on the genus distribution across the vertices of the graph). The extra kappa classes appearing in the k-twisted DR cycle formula are assimilated into our definition of the generalized boundary strata classes just as the psi classes were in the untwisted version. Acknowledgements I would like to thank G. Oberdieck for useful discussions when first trying to define generalized boundary strata classes. I would also like to thank the anonymous referee. I was supported by a fellowship from the Clay Mathematics Institute.

References 1. E. Arbarello, M. Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebraic Geom. 5(4), 705–749 (1996) 2. E. Clader, F. Janda, Pixton’s double ramification cycle relations. Geom. Topol. 22(2), 1069–1108 (2018) 3. C. Faber, R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring. Michigan Math. J. 48, 215–252 (2000). With an appendix by Don Zagier 4. C. Faber, R. Pandharipande, Relative maps and tautological classes. J. Eur. Math. Soc. 7(1), 13–49 (2005) 5. T. Graber, R. Pandharipande, Constructions of nontautological classes on moduli spaces of curves. Michigan Math. J. 51(1), 93–109 (2003) 6. F. Janda, R. Pandharipande, A. Pixton, D. Zvonkine, Double ramification cycles on the moduli spaces of curves. Publications mathématiques de l’IHÉS 125(1), 221–266 (2017) 7. R. Pandharipande, A. Pixton, D. Zvonkine, Relations on M g,n via 3 -spin structures. J. Am. Math. Soc. 28(1), 279–309 (2015)

Torsion Points of Sections of Lagrangian Torus Fibrations and the Chow Ring of Hyper-Kähler Manifolds Claire Voisin

Abstract Let φ : X → B be a Lagrangian fibration on a projective irreducible hyper-Kähler manifold. Let M ∈ Pic X be a line bundle whose restriction to the general fiber Xb of φ is topologically trivial. We prove that if the fibration is isotrivial or has maximal variation and X is of dimension ≤ 8, the set of points b such that the restriction M|Xb is torsion is dense in B. We give an application to the Chow ring of X, providing further evidence for Beauville’s weak splitting conjecture.

1 Introduction 1.1 The General Problem and Main Result Let X be a smooth algebraic variety over C, φ : X → B a projective morphism and let M ∈ Pic X be a line bundle which is topologically trivial on the fibers of φ. The line bundle M then determines an algebraic section νM of the torus fibration Pic0 (X0 /B 0 ) → B 0 (which we will call the normal function associated to M), where X0 → B 0 is the restriction of φ over the open set B 0 ⊂ B of regular values of φ. The group of such sections is called the Mordell-Weil group of the fibration. The problem we investigate in this paper is: Question 1.1 When do there exist points b ∈ B 0 where νM (b) is a torsion point? Are these points dense (for the usual or Zariski topology) in B?

C. Voisin () Collège de France, Paris, France ETH-ITS, Zürich, Switzerland e-mail: [email protected] © Springer Nature Switzerland AG 2018 J. A. Christophersen, K. Ranestad (eds.), Geometry of Moduli, Abel Symposia 14, https://doi.org/10.1007/978-3-319-94881-2_10

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Note that this question can be restated in terms of variations of mixed Hodge structures Et obtained as an extension 0 → H t → Et → Z → 0 of Z by a pure Hodge structure Ht of weight 1 (see for example [7, 8], [9]). Torsion points then correspond to those points t for which Et has a nonzero rational element which belongs to F 1 Et,C . In this setting, similar existence or density problems have been investigated in the study of the density of the Noether-Lefschetz locus (see [10, 32]), where variations of weight 2 Hodge structures are considered and special points t are those for which the Hodge structure Et acquires a Hodge class. More generally, the set of special points is the object of a vast literature related to the André-Oort conjecture and “unlikely intersection” theory [36], where the issue, however, is more focused on bounding special points than on their existence. It turns out that both problems are related to transversality questions so that there is in fact a certain overlap in the methods (see for example [12]). Question 1.1 is natural only when the codimension of the locus of torsion points in Pic0 (X/B), that is h1,0 (Xb ) = dim Pic0 (X/B) − dim B, is not greater than n = dim B, so that the section νM is expected to meet this locus. We will consider in this paper the case where h1,0 (Xb ) = dim B, and more precisely we will work in the following setting: X will be a projective hyper-Kähler manifold of dimension 2n and φ : X → B will be a Lagrangian fibration. According to Matsushita [23], any morphism φ : X → B with 0 < dim B < dim X and connected fibers provides such a Lagrangian fibration. The smooth fibers Xb are then abelian varieties of dimension n, which are in fact canonically polarized, by a result of Matsushita (see [25, Lemma 2.2] and Proposition 2.3). So in the following definition, the local period map could be replaced in our case by a moduli map from the base to a moduli space of abelian varieties. Definition 1.2 We will say that a smooth torus fibration X0 → B 0 has maximal variation if the (locally defined) classifying map B 0 → D, where D is the basis of the Kuranishi family of the fibres, is generically of maximal rank n = dim B. The fibration is said to satisfy Matsushita’s alternative if it has maximal variation or it is locally constant (the isotrivial case). Matsushita conjectured that this alternative holds for Lagrangian fibrations on projective hyper-Kähler manifolds. This conjecture was proved in [31] assuming that the Mumford-Tate group of the Hodge structure on the transcendental cohomology H 2 (X, Q)t r ⊂ H 2 (X, Q) is the full special orthogonal group of H 2 (X, Q)t r equipped with the BeauvilleBogomolov intersection form, and that b2 (X)t r := dim H 2 (X, Q)t r ≥ 5. In particular, it is satisfied by general deformations of X with fixed Picard lattice, assuming b2 (X)t r ≥ 5. Our main result in this paper is the following: Theorem 1.3 Let X, φ be as above, and let M ∈ Pic X restrict to a topologically trivial line bundle on the smooth fibers Xb of φ. Assume that either the fibration is

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locally isotrivial or dim X ≤ 8 and the variation is maximal. Then the set of points b ∈ B 0 such that νM (b) = M|Xb is a torsion point in Pic Xb is dense in B 0 for the usual topology. Applying the result of [31] mentioned above, we conclude: Corollary 1.4 Let X, φ, M be as above, so X is projective hyper-Kähler, φ is a Lagrangian fibration, M is topologically trivial on the fibers. Assume b2 (X) ≥ 8. Then the very general deformation Xt of X preserving L, M and an ample line bundle on X satisfies the conclusion of Theorem 1.3. The restriction to dimension ≤ 8 is certainly not essential here and we believe that the result is true in any dimension although the analysis seems to be very complicated in higher dimension. The restriction to dimension 8 appears in the analysis of the constraints on the infinitesimal variation of Hodge structures imposed by contradicting the conclusion of Theorem 1.3. In dimension 10 we completely classify the possible (although improbable) situation where the conclusion of Theorem 1.3 does not hold (see Proposition 3.8 proved in Sect. 7). In order to introduce some of the ingredients used in the proof of Theorem 1.3, we will start with the general case of an elliptic fibration, (and without any hyperKähler condition). One gets in this case the following result, which, although we could not find a reference, is presumably not new, but will serve as a toy example for the strategy used. Theorem 1.5 Let φ : X → B be an elliptic fibration, where B is a smooth projective variety and X is smooth projective, and let B 0 ⊂ B be the open set of regular values of φ. Let M be a line bundle on X which is of degree 0 on the fibers of φ. Then either the set of points b ∈ B 0 such that νM (b) = M|Xb is a torsion point in Pic Xb is dense in B 0 for the usual topology, or the restriction map H 1 (X, Q) → H 1 (Xb , Q) is surjective. In the second case, the fibration φ is locally isotrivial and the associated Jacobian fibration is rationally isogenous over B to the product J (Xb ) × B. Our approach to Theorem 1.3 is infinitesimal and completed by an analysis of the monodromy. It uses the easy Proposition 3.1 which says that the torsion points of a section ν of a family of complex tori are dense in the base B if the natural locally defined map fν : B → H1 (At0 , R) obtained from ν by a real analytic trivialization of the family of complex tori A → B, is generically submersive. This map is called the Betti map in [2, 12]. A key tool for our work is the following very useful result of André-Corvaja-Zannier [2]. Theorem 1.6 Let π : A → B be a family of abelian varieties of dimension d, where B is quasi-projective over C and dim B ≥ d. Let ν be an algebraic section of π. Assume (i) The family has no locally trivial subfamily. (ii) The multiples mν(B) are Zariski dense in A.

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Then if the real analytic map fν is nowhere of maximal rank, the variation of Hodge structure of weight 1 is degenerate in the following sense: For any b ∈ B, for any λ ∈ H 1,0(Ab ) the map ∇ λ : TB,b → H 0,1 (Ab ) is not of maximal rank, where TB,b is the tangent space to B at b. We refer to Sect. 4 for the definition of ∇. We will also for completeness sketch the proof of this result in that section.

1.2 An Application to the Chow Ring of Hyper-Kähler Fourfolds Let us explain one consequence of Theorem 1.3, which was our motivation for this work. Following the discovery made in [5] that a projective K3 surface S has a canonical 0-cycle oS with the property that for any two divisors D, D on S, the intersection D · D is a multiple of oS in CH0 (S), Beauville conjectured the following: Conjecture 1.7 (See [3]) Let X be a projective hyper-Kähler manifold. Then the cohomological cycle class map restricted to the subalgebra of CH(X)Q generated by divisor classes is injective. This conjecture has been proved in [35] for varieties of the form S [n] , where S is a K3 surface and n ≤ 2b2 (S)t r + 4, and in [15] for generalized Kummer varieties. It is also proved in [35] for Fano varieties of lines of cubic fourfolds, which are well-known to be irreducible hyper-Kähler fourfolds of K3[2] deformation type (see [4]). Finally, Conjecture 1.7 is proved by Riess [30] in the case of irreducible hyper-Kähler varieties of K3[n] or generalized Kummer deformation type admitting a Lagrangian fibration. Here the condition that X is a deformation of a punctual Hilbert scheme of a K3 surface guarantees, according to [24], that X satisfies the conjecture that any nef line bundle L on X which is isotropic for the BeauvilleBogomolov quadratic form q is the pull-back of a Q-line bundle on the basis B of a Lagrangian fibration on X. The application to Conjecture 1.7 we give in this paper concerns one natural relation in the cohomology ring of a hyper-Kähler manifold equipped with a Lagrangian fibration. Note that, according to [6], the set of polynomial relations between degree 2 cohomology classes on an irreducible hyper-Kähler 2n-fold X is generated as an ideal of Sym H 2 (X, R) by the relations α n+1 = 0 in H 2n+2 (X, R) if q(α) = 0.

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For the same reason, the cohomological relations between divisor classes d ∈ NS(X)R ⊂ H 2 (X, R) are generated by the relations d n+1 = 0 in H 2n+2 (X, R) if q(d) = 0.

(1)

In particular there are no relations in degree ≤ 2n. If X is any projective hyperKähler manifold, let l ∈ H 2 (X, Q) be such that q(l) = 0. One has then l n+1 = 0 in H 2n+2 (X, Q). If furthermore X admits a Lagrangian fibration such that l = c1 (L) and L is pulled-back from the base, then one clearly has Ln+1 = 0 in CH(X) and this is the starting point of Riess’ work [29, 30]. Observe next that by differentiation of (1) along the tangent space to Q : V (q) at l, namely l ⊥q , one gets the relations l n h = 0 in H 2n+2 (X, R) if q(l) = 0 and q(l, h) = 0.

(2)

Our application of Theorem 1.3 is the following result, proving that if l = c1 (L) as above comes from a Lagrangian fibration, d = c1 (D) is a divisor class with q(l, d) = 0, and the dimension is ≤ 8, then (2) already holds in CH(X)Q . Theorem 1.8 Let φ : X → B be a Lagrangian fibration on a projective irreducible hyper-Kähler manifold of dimension 2n ≤ 8 with b2 (X) ≥ 8 and let L generate φ ∗ Pic B. Let furthermore D ∈ Pic X satisfy the property that Ln ·D is cohomologous to 0 on X. Then Ln · D = 0 in CHn+1 (X)Q . We refer to [20] for further applications of Theorem 1.8 to Conjecture 1.7. We close this introduction by mentioning mentioning other applications other potential applications of the study of Question 1.1 in various geometric contexts. Here is an example: Let S be a K3 surface. Huybrechts [17] defined a constant cycle curve C ⊂ S to be a curve all of whose points are rationally equivalent in S. All points of C are then rationally equivalent to the canonical cycle oS . Rational curves in S are constant cycles curves, but there are many other constant cycles curves, some of which can be constructed as follows: Choose an ample linear system |L| on S and let C ⊂ |L| × S be the universal curve. Over the regular locus |L|0 , we have the Jacobian fibration J → |L|0 and we get a section ν of the pull-back JC on C by defining ν(c, C) = albC (dc − L|C ), where d = L2 so that dc − L|C has degree 0 on C. The locus where this section is torsion is expected to be 1-dimensional and to project via the natural map C → S to a countable union of constant cycles curves. An analysis of Question 1.1 is necessary here to make this expectation into a valid statement. Another related example is as follows: Let now |L1 |, |L2 | be two linear systems on # the K3 surface S, and let N = L1 · L2 . Choose integers w1 , . . . , wN such that i wi = 0. Over the open set (|L1 | × |L2 |)0 where the curves C1 , C2 are smooth and meet transversally, there is an étale cover Z → (|L1 | × |L2 |)0 parameterizing orderings of the set C1 ∩ C2 and

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two sections ν1 : Z → J1,Z , ν2 : Z → J2,Z , where J1,Z , J2,Z are the pull-backs to Z of the Jacobian fibrations on |L1 |, resp. |L2 |. They are defined by ν1 (z) = albC1,z (

 i

wi xi,z ), ν2 (z) = albC2,z (



wi xi,z ),

i

where z ∈ Z parameterizes the curves C1,z , C2,z and the ordering C1,z ∩ C2,z = (x1,z , . . . , xN,z ) of their intersections. The vanishing (or torsion) locus of (ν1 , ν2 ) is expected to be 0-dimensional. It provides a geometric way of producing elements of the higher Chow group CH2 (S, 1) (see [11, 26, 34]). The paper is organized as follows. In the very short Sect. 2, we will show how Theorem 1.8 follows from Theorem 1.3. We will prove Theorem 1.5 in Sect. 3. Section 4 will be devoted to sketching the proof of the André-Corvaja-Zannier theorem. The proof of Theorem 1.3 will be given in Sect. 5 in the non-isotrivial case and in Sect. 6 in the isotrivial case. Sections 5.1, 6.1 and 6.2 present in detail the local analysis of the nontransversality of a Lagrangian section of a Lagrangian torus fibration and show how it is related to a degenerate real Monge-Ampère equation. In Sect. 7, we will analyze the case of a 10-dimensional Lagrangian fibration and prove Proposition 3.8. Thanks I thank Sébastien Boucksom and Jean-Pierre Demailly for useful discussions concerning the degenerate Monge-Ampère equation. I also thank Pietro Corvaja and Umberto Zannier for introducing me to the number theorist terminology of Betti coordinates, for providing a guide through the existing literature, for communicating their paper [2] and for explaining to me their results. The present paper is a completely new version of the paper arXiv:1603.04320, and the use of their main Theorem 1.6 led me to a much improved result.

2 Application of Theorem 1.3 to the Beauville Conjecture We prove in this section Theorem 1.8 assuming Theorem 1.3. Let φ : X → B be as in the introduction, with dim X = 2n ≤ 8 and b2 (X) ≥ 8 and let L be a holomorphic line bundle on X generating φ ∗ Pic B (see [23]). We want to prove that for any line bundle D on X such that D · Ln is cohomologous to 0 on X, then D · Ln is rationally equivalent to 0 on X modulo torsion. Let us consider a local universal family X → T of deformations of X preserving the line bundles L, D and an ample line bundle on X. The very general fiber Xt of this family is projective and admits the deformed line bundles Dt and Lt . It also admits a Lagrangian fibration associated to Lt (see [25], and also [31] if the base of

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the fibration is not smooth). The condition that Dt · Lnt is rationally equivalent to 0 on Xt modulo torsion is satisfied on a countable union of closed algebraic subsets of T , so if we prove it is satisfied at the very general point t ∈ T , it will be also satisfied for any t ∈ T , hence for X. As b2 (X) ≥ 8, we can apply Corollary 1.4 to the very general deformation (Xt , φt , Lt , Dt ). It thus suffices to prove the result when (X, φ, D) satisfies the conclusion of Theorem 1.3, which we assume from now on. In the Chow ring CH(X)Q , the fibers Xb , for b ∈ B 0 , are all rationally equivalent (if B is smooth, then B ∼ = Pn by [18], so this is obvious; if B is not smooth, we refer to [20] for a proof of this statement). It follows that we have for any b ∈ B Ln = μXb in CHn (X)Q ,

(3)

for some nonzero μ ∈ Z. Let b ∈ B be a general point. The kernel of the map [Xb ]∪ =

1 c1 (L)n ∪ : H 2 (X, Q) → H 2n+2 (X, Q) μ

is equal to the kernel of the restriction map H 2 (X, Q) → H 2 (Xb , Q). This is a general fact proved in [33, Lemme 1.5 and Remarque 1.6]: Lemma 2.1 (See [33]) Let j : Y → W be a generically finite morphism, where Y and W are smooth projective varieties (in particular connected), and let dim W − dim Y = k, dim W = m. Let α := j∗ [Y ]f und ∈ H 2k (W, Q). Then the two Q-vector subspaces K1 := Ker (j ∗ : H 2 (W, Q) → H 2 (Y, Q)), K2 := Ker (α∪ : H 2 (W, Q) → H 2k+2 (W, Q))

of H 2 (W, Q) are equal. Let now D ∈ Pic X such that D·Ln is cohomologous to 0. Then c1 (D)∪[Xb ] = 0 by (3), and thus c1 (D)|Xb = 0 in H 2 (Xb , Q), hence also in H 2 (Xb , Z). As we assumed that (X, φ, D) satisfies the conclusion of Theorem 1.3, there exist points in B 0 such that D|Xb is a torsion point in Pic0 (Xb ), so that D · Xb is a torsion cycle in CHn+1 (X). This implies by (3) that D ·Ln vanishes in CHn+1 (X)Q , which proves Theorem 1.8. Remark 2.2 In our specific case, Lemma 2.1 also follows directly from Proposition 2.3 below, that will be also used later on. Proposition 2.3 (Matsushita [25]) In the situation above, the restriction map H 2 (X, Q) → H 2 (Xb , Q) has rank 1. Indeed, K1 is always included in K2 , and in the other direction, if [Xb ] ∪ D is cohomologous to 0 in X, D|Xb cannot be ample, hence it must have trivial first Chern class by Proposition 2.3.

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3 A Toy Case: Elliptic Fibrations We will use the same notation φ : X0 → B 0 for the restriction to B 0 of the elliptic fibration φ : X → B. The associated Jacobian fibration J → B 0 is a complex manifold which is a fibration into complex tori, and its sheaf of holomorphic sections J is described in complex analytic terms as the quotient J = R 1 φ∗ OX0 /R 1 φ∗ Z = R 1 φ∗ C ⊗ OB /(H 1,0 ⊕ R 1 φ∗ Z) = H 0,1 /R 1 φ∗ Z, (4) where H 1,0 = R 0 φ∗ ΩX0 /B 0 , H 0,1 := R 1 φ∗ OX0 = (R 1 φ∗ C ⊗C OB 0 )/H 1,0 . Formula (4) describes J as a holomorphic torus fibration. As a C ω real torus fibration, however, the right formula is JR = HR1 /R 1 φ∗ Z, HR1 := R 1 φ∗ R ⊗ CBω0 ,R

(5)

which uses the natural fiberwise isomorphisms H 1 (Xb , R) ∼ = H 1 (Xb , C)/H 1,0(Xb ) ∼ = H 1 (Xb , OXb )

(6)

globalizing into the CBω0 ,R sheaf isomorphism HR1 ∼ = R 1 φ∗ OX0 ⊗OB CBω0 ,C ∼ = H 0,1 ⊗OB CBω0 ,C .

(7)

Trivializing the locally constant sheaves R 1 φ∗ Z, R 1 φ∗ R on simply connected open sets U ⊂ B, (5) is the counterpart, at the level of sheaves of sections, of a local real analytic trivialization over U JU ∼ = U × (H 1 (X0 , Z) ⊗ R/Z),

(8)

where 0 is a given point of U . Let now ν be a holomorphic section of J over U . Using the trivialization (8), the section ν gives a differentiable (in fact real analytic) map fν : U → H 1 (X0 , Z) ⊗ R/Z = J (X0 ), (which is called the Betti map in [12]) and we will use the following easy criterion: Proposition 3.1 (i) For a section ν of a torus fibration with local associated map fν as above, the points x of U where ν(x) is of torsion are dense in U for the usual topology if the differential dfν is surjective at some point b ∈ U .

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(ii) In the situation of an elliptic fibration over a 1-dimensional base, the differential dfν is surjective if and only if it is nonzero. Proof (i) Observe first that fν being real analytic, if its differential is surjective at some point b ∈ U , it is surjective on a dense set of points of U , hence it suffices to prove density near a point b where the differential dfν,b is surjective. If the differential dfν,b is surjective, fν is an open map in a neighborhood Ub of b. As the torsion points are dense in J (Xb0 ), their preimages under fν are then dense in Ub . (ii) This is a consequence of the following fact: Lemma 3.2 The kernel of the differential dfν,b : TU,b → TJ (X0 ) = H 1 (X0 , R) is a complex vector subspace of TU,b . Proof Let us prove more generally that the fibers of fν are analytic subschemes of U . Let α ∈ J (X0 ) = H 1 (X0 , R)/H 1 (X0 , Z) and let α˜ ∈ H 1 (X0 , R) be a lifting of α (such a lift is defined up to the addition of an element β of H 1 (X0 , Z)). The class α˜ extends as a constant section also denoted α˜ of the sheaf R 1 φ∗ R on U , which induces a holomorphic section α˜ 0,1 of the sheaf H 0,1 = R 1 φ∗0 OX0 on U . On the other hand, the holomorphic section ν of J lifts to a holomorphic section η of H 0,1 and it is clear that fν−1 (α) = {b ∈ U, α˜ b0,1 = ηb in H 0,1 (Xb )/H 1 (Xb , Z)}. It follows that fν−1 (α) is the countable locally finite union of the closed analytic subsets defined as zero sets of the holomorphic sections α˜ 0,1 − η − β 0,1 ∈ Γ (U, H 0,1 ) of the bundle H 0,1 , over all sections β of HZ1 on U . In the situation of (ii), the base U and the fiber J (X0 ) are both of real dimension 2 and Lemma 3.2 implies that the kernel of dfν,b is of real dimension 0 or 2. So either dfν,b = 0 or dfν,b is surjective. Proof (Proof of Theorem 1.5) Let φ : X0 → B 0 be our elliptic fibration and ν a section of J (X0 /B). Assume that the points b of B where ν(b) is a torsion point are not dense for the usual topology of B. Then by Proposition 3.1, it follows that dfν vanishes everywhere on any open set U of B 0 where it is defined. Coming back to the sheaf theoretic language, this means equivalently that ν, seen as a section of HR1 /HZ1 via the isomorphism (5), is locally constant, or that our section ν ∈ Γ (B 0 , H 0,1 /HZ1 ) comes from a locally constant section ν˜ R ∈ Γ (B 0 , HR1 /HZ1 ). We are thus in position to apply Manin’s theorem on the kernel [22] and conclude. For further use, we give the complete argument here. Note that, by assumption, ν˜ R is not of torsion and thus, fixing a base point 0 ∈ B, corresponds to an element α0 ∈

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H 1 (X0 , R) which is not rational but has the property that for any γ ∈ π1 (B 0 , 0), ρ(γ )(α0 ) − α0 ∈ H 1 (X0 , Z),

(9)

where ρ : π1 (B 0 , 0) → Aut H 1 (X0 , Z) denotes the monodromy representation of the smooth fibration φ : X0 → B 0 . We use the following easy lemma. Lemma 3.3 Let ρ : Γ → Aut VQ be a finite dimensional rational representation of a group Γ . Then if the invariant space VQinv is trivial, the set {v ∈ VR , ρ(γ )(v) − v ∈ VQ for any γ ∈ Γ } is equal to VQ . As the monodromy representation is rational, this lemma tells us that the set of classes α ∈ H 1 (X0 , R) satisfying property (9) contains a nonrational class if and only if the set H 1 (X0 , Q)inv := {α ∈ H 1 (X0 , Q), ρ(γ )(α) − α = 0, ∀γ ∈ π1 (B 0 , 0)} of monodromy invariant elements is nontrivial. By Deligne’s invariant cycles theorem [13], it then follows from the existence of α0 that the restriction map H 1 (X, Q) → H 1 (X0 , Q) is nontrivial, hence surjective since this is a morphism of Hodge structures of weight 1 and the right hand side has dimension 2. The conclusion that X is rationally isogenous to J (X0 ) × B is then immediate since X is rationally isogenous to a projective completion of the Jacobian fibration J (X0 /B 0 ) and the later is isogenous to J (X0 ) × B 0 if the restriction map H 1 (X, Q) → H 1 (X0 , Q) is surjective.

3.1 Some Examples with Higher Dimensional Fiber Dimension Looking at Theorem 1.5, one may wonder what makes the generalization to higher fiber dimension problematic. Let us give some examples explaining the main difficulties encountered: Consider more generally any complex torus fibration φA : A → B with a section νA . The torus fibration is canonically isomorphic, as a real torus fibration, to the locally constant fibration H1,R,φA /H1,Z,φA , where H1,R,φA := (R 1 φA∗ R)∗ . A section νA thus admits local liftings ν˜ A,R which are C ∞ (in fact real analytic) sections of the flat vector bundle associated with the local system H1,R,φA .

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Example 3.4 Assume that the local lifts ν˜ A,R of the section νA are constant sections ν˜ A,R ∈ Γ (H1,R,φA ). Then if ν˜ A,R is not rational, that is, does not belong to Γ (H1,Q,φA ), νA has no torsion point. This case, which corresponds to the situation where the local maps fνA of the previous section are constant, can be in general excluded by a monodromy argument. The following example is slightly more subtle: Example 3.5 Assume that the torus fibration φ : A → B is a fibered product A = A ×B A

, where φ : A → B, φ

: A

→ B are torus fibrations. Choose a section νA

: B → A

of φ

which is as in Example 3.4, that is, locally lifts to a constant nonrational section of (R 1 φ∗

R)∗ . Then for any section νA : B → A of φ , the section νA = (νA , νA

) of φ does not have any torsion point. Another slightly more involved situation where we can avoid torsion points is as follows: Recall from the introduction that, if dim B < g := dim Ab , a “generic” section of the abelian scheme φA : A → B avoids the torsion points. Of course, for a general family A → B, there might be no non-trivial section, but there are always multisections and they can be chosen to pass through any point with arbitrary differential. By “generic” we mean generic section of a base-changed family A → B . The next example exhibiting nontransversality is as follows: Example 3.6 Choose A1 → B1 with dim B1 < g1 := dim A1,b and a section ν1 : B1 → A1 with no torsion points. For any A2 → B2 , such that dim B1 + dim B2 = g1 +g2 , and for any section ν2 : B2 → A2 , the section (ν1 , ν2 ) : B1 ×B2 → A1 ×A2 of the product family A1 × A2 → B1 × B2 has no torsion point. Let us conclude with the following abstract situation where we do not have transversality of the normal function (or rather of its local real analytic representation as in Sect. 3, so we do not expect the normal function to have torsion points. Example 3.7 Assume the abelian scheme A → B satisfies dim B = g = dim Ab and has the following property: B admits a foliation given by an algebraic distribution which is analytically integrable with holomorphic leaves Ft of dimension d, such that along the leaves, the real variation of Hodge structures on H1|Ft splits as H1,R|Ft = LR,t ⊕ L R,t , with rank L R = 2d , and d < d. Then if furthermore, the normal function ν : B → A or rather its local real analytic lift ν˜ R decomposes along the leaves as νR|Ft = νL,t + νL ,t with νL,t locally constant, then νR is never of maximal rank. Indeed, the differential dνR cannot be injective at any point since its restriction to the tangent space of the leaf is equal to dνL ,t : TFt ,R → L R,t , the two spaces being of dimensions 2d > 2d . This situation seems to be improbable if the LR,t furthermore varies with t (that is, does not come from a local system on B). However this is the only case that

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we could not exclude for normal sections associated with divisors on hyper-Kähler manifolds of dimension 10. Let X → B be a projective hyper-Kähler manifold of dimension 10 equipped with a Lagrangian fibration with maximal variation, and let M ∈ Pic X be a divisor which is cohomologous to 0 on fibers. The following result will be proved in Sect. 7: Proposition 3.8 (i) If the torsion points of νM : B 0 → X0 are not dense in B for the usual topology, a Zariski dense open set B 0 has a foliation with 3-dimensional leaves Ft , and the variation of Hodge structure along the leaves decomposes as H1,R|Ft = LR,t ⊕ L R,t where rank LR,t = 6, rank L R,t = 4. Furthermore, the real variation of Hodge structure on LR,t is constant along Ft . (ii) Along each leaf Ft , the normal function νM has the L-component ν˜ M,R,L of its real lift constant.

4 The André-Corvaja-Zannier Theorem We describe in this section following [2] the proof of Theorem 1.6. The reason for doing so is not only the fact that this result is important and the arguments beautiful, but also the fact that their paper is written with notations that are not familiar to Hodge theorists. We first comment on the meaning of the conclusion of the theorem and the notation used there: consider the local system H1,Z = (R 1 π∗ Z)∗ on B. It generates the flat holomorphic vector bundle H1 := H1,Z ⊗ OB , which carries the Hodge filtration H1,0 ⊂ H1 with quotient H0,1 . The Griffiths ∇ map is defined as the composite ∇

H1,0 → H1 ⊗ ΩB → H0,1 ⊗ ΩB , where ∇ is the Gauss-Manin connection. It is OB -linear, hence induces, for each b ∈ B, λ ∈ H1,0,b , a linear map: ∇ λ : TB,b → H0,1,b . A crucial ingredient in the proof of Theorem 1.6 is the following result due to André [1]: In the situation of the theorem, let Γ0 ⊂ π1 (B, b) be the subgroup acting trivially on H 1 (Ab , Z). On the cover BΓ0 → B equipped with a point b˜ over b ∈ B, the base-changed fibration AΓ0 → BΓ0 has a natural globally defined real analytic

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trivialization AΓ0 ∼ = B × H1 (Ab˜ , R)/H1 (Ab˜ , Z), so that the section ν becomes a well-defined real analytic map fν : BΓ0 → H1 (Ab˜ , R)/H1 (Ab˜ , Z). Theorem 4.1 (André [1]) Under the assumptions (i) and (ii), the image of the monodromy Γ0 → H1 (Ab˜ , Z) of fν is Zariski dense in H1 (Ab˜ , C). Proof (Proof of Theorem 1.6) We work on BΓ0 . The real analytic map fν is constructed as follows: The holomorphic section ν on B is a section of the sheaf H0,1 /H1,Z . Choose a local lift ν0,1 ∈ Γ (H0,1 ). Due to Hodge symmetry, the real analytic flat vector bundle H1,R is isomorphic to H0,1 as a real analytic vector bundle. In this way, the holomorphic section ν0,1 of H0,1 provides a real analytic section ν˜R of H1,R . We need to describe more explicitly how ν˜R is deduced from ν0,1 . Choose a local holomorphic lift ν˜ hol of ν to a holomorphic section of the bundle H1 . We use now the real analytic decomposition H1,an ∼ = H1,0,an ⊕ H0,1,an ,

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with H0,1 = H1,0 . Here the index “an” indicates that we consider the associated real analytic complex vector bundle. We can thus write ν˜ hol − ν˜ hol = η1 − η1 for some real analytic section η1 of H1,0 . Then we conclude that ν˜ hol − η1 = ν˜ hol − η1

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is real and maps to ν0,1 via the quotient map H1,an → H0,1,an . Thus ν˜ hol − η1 gives the desired section ν˜ R of H1,R,an (which is then made into a real analytic map fν : B → H1 (Ab , R) by local flat trivialization of H1,R,an ). Note that it is clearly independent of the choice of holomorphic lifting ν˜ hol of ν0,1 . Our assumption is that fν is nowhere of maximal rank. What is unpleasant about fν is its real analytic, as opposed to holomorphic, character. The first step in the André-Corvaja-Zannier proof is the separation of holomorphic and antiholomorphic variables so as to make this assumption into a holomorphic equation. Let us work locally on B × B (later on, we will rather consider BΓ0 × BΓ0 ). Here B is B equipped with its conjugate complex structure, for which the holomorphic functions are the complex conjugates of holomorphic functions on B. Over B we have the holomorphic bundle H1,0 and this provides us with two holomorphic vector bundles K1 := pr1∗ H1,0 , K2 := pr2∗ H1,0 on B × B. A neighborhood for the Euclidean topology of the diagonal of B in B × B retracts onto ΔB , hence carries a local system extending the local system H1,C on B ∼ = ΔB , with associated flat holomorphic bundle H1 . This vector bundle

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is isomorphic (near ΔB ) to both p1∗ H1 and p2∗ H1 . The inclusions of holomorphic subbundles H1,0 ⊂ H1 , resp. H1,0 ⊂ H1 on B, resp. B gives on B × B two holomorphic vector subbundles K1 ⊂ H1 , K2 ⊂ H1 , which restricted to the diagonal produce (10). Thus we have H1 = K1 ⊕ K2

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near the diagonal of B in B × B. We now produce a holomorphic version of ν˜ R on B × B near the diagonal as follows: starting from local lifts ν˜ hol of ν0,1 to a holomorphic section of H1 on B, we consider ν˜ 1 := pr1∗ ν˜hol , ν˜ 2 = pr2∗ ν˜ hol as holomorphic sections of H1 defined on B × B near the diagonal and write, using (12), ν˜1 − ν˜ 2 = η1 − η2 , where η1 , η2 are holomorphic sections of K1 , K2 respectively. We then consider the holomorphic section ν˜ 1 − η1 of H1 and observe that its restriction to the diagonal of B is exactly our real lifting ν˜R . Similarly, the locally defined holomorphic map Fν : B × B → H 1 (Ab , C) deduced from ν˜ 1 − η1 by locally trivializing the flat vector bundle H1 , restricts to fν : B → H 1 (Ab , R) on the diagonal ΔB . One easily shows (see [2, Lemma 4.2.1]) that Fν : B × B → H 1 (Ab , C) is generically of maximal rank if and only if fν : B → H 1 (Ab , R) is generically of maximal rank. Thus the assumption of the theorem is that Fν is nowhere of maximal rank where it is defined, namely near the diagonal of B. It is important to observe that ν˜ 1 − η1 and Fν in fact do not depend on the choice of lift ν˜ of ν0,1 . Indeed, if we add to ν˜ a section λ of H1,0 without changing ν˜ , then we have ν˜1 = ν˜ 1 + λ1 , with λ1 = pr1∗ λ, and ν˜2 = ν˜ 2 , so that ν˜ 1 − ν˜ 2 = ν˜ 1 − ν˜ 2 + λ1 and η1 = η1 + λ1 . Thus ν˜1 − η1 = ν˜ 1 − η1 . Similarly, if we change now ν˜ by adding to it a section λ of H0,1 , then we do not change ν˜ 1 and we have ν˜2 = ν˜ 2 + λ2 where λ2 = pr2∗ λ. Then η1 = η1 and η2 = η2 + λ2 , so finally ν˜ 1 − η1 = ν˜1 − η1 . In order to use the monodromy Theorem 4.1, we need to make the above construction more global, and this can be done by working on B := BΓ0 × BΓ0 . On BΓ0 , the local system H1 is trivial, and thus we have a global isomorphism pr1∗ H1 ∼ = ∗ ∗ ∗ pr2 H1 = H , so that K1 = pr1 H1,0 and K2 = pr2 H1,0 are holomorphic subbundles of the trivial bundle H := H ⊗ OB . On a dense analytic-Zariski open set B0 of B , we have H ∼ = H /K1 . We thus = K1 ⊕ K2 . Furthermore pr1∗ H0,1 ∼ get a multivalued holomorphic map on B0

Fν : B0 → H1 (Ab˜ , C)

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which coincides with the previously defined map near the diagonal of BΓ0 . The fact that Fν is multivalued follows from the fact that it is well defined once the lifts ν0,1 , ν0,1 are chosen, but it depends on these lifts. We now use Theorem 4.1 on BΓ0 . It says that the couple (ν0,1 , ν0,1 ) (or their lifts ν˜ , ν˜ ) changes under monodromy along B (or B0 ) by the addition of couples (u, v) ∈ H1 (Ab˜ , Z)2 with u, v in a Zariski dense subset of H1 (Ab˜ , Z)2 ⊂ H1 (Ab˜ , C)2 . Write now ν˜u = ν˜ + u, ν˜ v = ν˜ + v. Then, using (12), we can write u − v = λ1 − λ2 on B0 , where λ1 is a holomorphic section of K1 and λ2 is a holomorphic section of K2 , and we get pr1∗ ν˜u − pr2∗ ν˜ v = pr1∗ ν˜ 0,1 − pr2∗ ν˜ 0,1 + u − v = pr1∗ ν˜ 0,1 − pr2∗ ν˜ 0,1 + λ1 − λ2 = η1 − η2 + λ1 − λ2 , so that ν˜ 1,u,v = ν˜ 1 + u and η1,u,v = η1 + λ1 . According to the recipe described above, the map Fν thus becomes Fν + u − λ1 . A translation by u does not change the differential of Fν , so we can do here u = 0 and choose v in a Zariski dense set of H 1 (Ab˜ , Z). As the holomorphic map Fν is nowhere of maximal rank, we conclude that for any v in a Zariski dense set of H 1 (Ab˜ , Z) ⊂ H 1 (Ab˜ , C), the holomorphic map Fν − λ1 : B0 → H 1 (Ab˜ , C) is nowhere of maximal rank, where λ1 is the holomorphic section of K1 ⊂ H1 defined by v = λ1 − λ2 . We now apply the following easy lemma: for any μ ∈ H 1 (Ab˜ , C), we can write as before, using (12) μ = μ1 + μ2 as sections of H on B0 , with μi a holomorphic section of Ki , i = 1, 2. We can see μ1 as a holomorphic map B0 → H = H1 (Ab˜ , C). Lemma 4.2 Fix b ∈ B0 . Then the set of μ ∈ H1 (Ab , C) such that the holomorphic map Fν − μ1 is not of maximal rank at b is Zariski closed in H1 (Ab , C). Proof Indeed, the assignment μ → μ1 is C-linear in μ, hence for fixed b, μ1 , dμ1,b depend C-linearly on μ, which concludes the proof. Lemma 4.2 and the fact that v above can be taken in a Zariski dense set of H 1 (Ab˜ , Z) ⊂ H 1 (Ab˜ , C) allow now to conclude that for any b ∈ B and any μ ∈ H1 (Ab , C), the holomorphic map Fν − μ1 is not of maximal rank at b so that

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μ1 is not of maximal rank at b . The proof of Theorem 1.6 concludes now with the following lemma: Lemma 4.3 Assume that for any b ∈ B0 , and any μ ∈ H1 (Ab , C), the differential dμ1 : TB ,b → H1 (Ab , C) is not of maximal rank. Then for any b ∈ B, and any λ ∈ H1,0(Ab ) the map ∇ λ : TB,b → H0,1,b is not of maximal rank. Proof The subbundle K1 ⊂ H , where H is trivial, has a variation δ1 : K1 → K2 ⊗ ΩB and similarly for K2 . Along B ⊂ B × B, δ1 = ∇ and δ2 is its complex conjugate. The differential dμ1 is computed as follows: we have dμ = 0 = dμ1 + dμ2 ,

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with dμ1 = d1 μ1 + δ1 (μ1 ), dμ2 = d2 μ2 + δ2 (μ2 ) for some differentials di μi ∈ ΩB ⊗ Ki . It follows from (14) that d1 μ1 + δ2 (μ2 ) = 0, d2 μ2 + δ1 (μ1 ) = 0, so that dμ1 = −δ2 (μ2 ) + δ1 (μ1 ).

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Suppose there exist b ∈ B and λ ∈ H 1,0(Ab ) such that the map ∇(λ) : TB,b → H0,1,b is of maximal rank. Let μ = λ + λ. Equation (15) at b ∈ B ⊂ B × B gives then dμ1 = −∇(λ) + ∇(λ) and this sum is the direct sum of the two isomorphisms ∇(λ) : TB ∼ = H0,1 (Ab ), −∇(λ) : TB ∼ = H1,0 (Ab ), hence it is an isomorphism dμ1 : TB ,b = TB ⊕ TB ∼ = H1 (Ab , C), contradicting our assumption. This concludes the proof of Theorem 1.6. We finish this section by observing that the proof given above allows to prove in fact a statement slightly stronger than Theorem 1.6, namely the following variant, for which we introduce a notation: associated to our normal function ν, which is a section of the sheaf J = H0,1 /H1,Z and given a local lift of ν to a holomorphic section ν0,1 of H0,1 , we get an affine subbundle H1,0,ν of H1 = H1,Z ⊗ OB defined as H1,0,ν := q −1 (ν0,1 ),

(16)

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where q : H1 → H0,1 is the quotient map. Composing the inclusion H1,0,ν ⊂ H1 of the affine subbundle H1,0,ν with a local flat trivialization Φ : H1 → H1,b,C , we get a holomorphic map Φν : H1,0,ν → H1,b,C .

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Theorem 4.4 Let π : A → B be a family of abelian varieties of dimension d over C, where B is quasi-projective and dim B ≥ d. Let ν be an algebraic section of π. Assume (i) The family has no locally trivial subfamily. (ii) The multiples mν(B) are Zariski dense in A. Then if the associated real analytic map fν is nowhere of maximal rank, the map Φν is nowhere submersive. Note that this strengthens Theorem 1.6 since the conclusion of Theorem 1.6 is the fact that the holomorphic map Φ1,0 := Φ|H1,0 : H1,0 → H1,b,C is nowhere of maximal rank. Choosing a holomorphic lift ν˜ of ν in H1 , the map Φν of (17) identifies to Φ ◦ ν˜ + Φ1,0 : H1,0 → H1,b,C and it is clear, by passing to the linear part, that if Φ ◦ ν˜ + Φ0,1 is nowhere of maximal rank on H1,0 , so is Φ1,0 .

5 Lagrangian Fibrations on Hyper-Kähler Manifolds; The Non-isotrivial Case of Theorem 1.3 5.1

Local Structure Results

We recall in this section, following [14] or [16], the local data determining a holomorphic Lagrangian polarized torus fibration. The holomorphic torus fibration φ : A → U is determined by a weight 1 (or rather −1) variation of Hodge structure, that is, the data of a local system H1,Z = (R 1 φ∗ Z)∗ and a holomorphic subbundle H1,0 ⊂ H1 := H1,Z ⊗ OU determining a weight 1 Hodge structure at any point of U . The sheaf A of holomorphic sections of A identifies to H0,1 /H1,Z , H0,1 := H1 /H1,0 .

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The holomorphic 2-form σA on A for which the fibration φ : A → U is a Lagrangian fibration provides by interior product an isomorphism of vector bundles TU = R 0 φ∗ (TA /TA/U ) ∼ = R 0 φ∗ ΩA/U ,

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which by dualization provides an isomorphism (which is canonical, given the choice of σA ) H0,1 ∼ = ΩU .

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Using the isomorphism of (20), the surjective map of holomorphic vector bundles H1 → H0,1 is thus given, on any simply connected open set U where the flat vector bundle H1 is trivialized, by the evaluation morphism of 2n holomorphic 1-forms αi on U , which must have the property that their real parts Re αi form a basis of ΩU ,R at any point. We have (see [14]): Lemma 5.1 The forms αi are closed on U , hence we have, up to shrinking our local open set U if necessary, αi = dfi , where the fi ’s are holomorphic and defined up to a constant. Proof The proof will use a different description of the forms αi . For any locally 2g−1 constant section γ of H1,C ∼ = HC , g = dim Ab , we get using the closed 2form σA a closed 1-form φ∗ (γ ∪ σA ) on U (when γ is a section of H1,Z , γ is a combination of classes γi of oriented circle bundles Γi ⊂ A, and φ∗ (γ ∪ σA ) is defined as (φ|Γi )∗ (σA|Γi )). We conclude observing that for any locally constant section γ of H1,C on an open set U of U , with induced section γ0,1 of H0,1 , providing a holomorphic form αγ0,1 via the isomorphism (20), we have αγ0,1 = φ∗ (γ ∪ σA ) in Γ (U , ΩU ), thus proving that the forms αi are closed. If we now choose the αi to form a basis of H1,R , the corresponding 2n holomorphic 1-forms on U have the properties that at any point b ∈ U , the forms αi,b are independent over R. Another way to say it is that the functions Re fi give local real coordinates on U . Remark 5.2 By definition, the functions Re fi , which are defined up to a constant and depend only on the choice of σA and of local basis αi of H1,R , provide a local real analytic identification U∼ = H 1 (Ab0 , R).

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Globally, they provide an affine flat structure on U . The last piece of information we need concerning the torus fibration φ : A → U is the polarization ω on the fibers. It is given by a monodromy invariant skewsymmetric pairing * , , on H1,R (we do not need here the fact that the polarization

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is integral), which has to polarize the Hodge structure at any point, so that for any b∈U * , , = 0 on H1,0,b ⊂ H1,C,b

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i*α, α, > 0, ∀0 = α ∈ H1,0,b . 1 This can be viewed as a nondegenerate skew-symmetric form ω ∈ 2 H 1 (Ab0 , R) which produces via the diffeomorphism (21) a closed 2-form ω∗ on the open set U of the trivialization introduced above. The 2-form ω∗ is constant in the affine coordinates introduced above. We have the following lemma whose proof is a formal computation (see [16]): Lemma 5.3 The Hodge-Riemann bilinear relations (22) satisfied by * , , and the Hodge structure on H1 (Ab , R) at any point b ∈ U are equivalent to the fact that ω∗ is a Kähler form on U . As a corollary of the above description, we get the following result due to Donagi and Markman [14]: Theorem 5.4 (Donagi-Markman) Let X → B be a Lagrangian fibration. Then locally on Breg , there exist a holomorphic function g and holomorphic coordinates z1 , . . . , zn such that the infinitesimal variation of Hodge structures TB,b → Hom (H1,0,b , H0,1,b ) at b ∈ Breg is induced by the cubic form g (3) defined by the ∗ third partial derivatives of g, using the identifications TB,b ∼ . = H1,0,b ∼ = H0,1,b Proof Indeed, let fi be the local holomorphic functions appearing above. The indices correspond to a local flat basis e1 , . . . , e2n of H1,Z . We can assume that the basis is chosen so that *e1 , . . . , en , is totally isotropic for the pairing giving the polarization. We choose for coordinates z1 = f1 , . . . , zn = fn . We have dfn+i =

n 

gij dzj ,

j =1

with gij =

∂fn+i ∂zj ,

so that the kernel of the evaluation map C2n ⊗ OB → ΩB

# is generated by en+i − nj=1 gij ej for j = 1, . . . , n. As the subspace generated by these elements is totally isotropic for the pairing, we get that gij = gj i . Thus we ∂f have ∂f∂zn+i = ∂zn+j for any i, j , and this implies that locally there is a holomorphic j i function g such that fn+i =

∂g ∂zi .

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We finally compute the infinitesimal # variation of Hodge structures by differentiating the generators ei := en+i − nj=1 gij ej of H1,0 : we get  ∂gij  ∂ei

∂ 3g =− ej = − ej . ∂zk ∂zk ∂zi ∂zj ∂zk j

j

This proves the claim as the basis ei , i ≥ n + 1 is dual to the basis ei for i ≤ n, which itself corresponds to the basis ∂z∂ i of TB,b .

5.2 Proof of Theorem 1.3 When dim X ≤ 8 and the Variation is Maximal We start with the following easy lemma which holds without any assumption on the variation: Lemma 5.5 Let φ : X → B be a Lagrangian fibration on a hyper-Kähler manifold. Then there is no proper nontrivial real subvariation of Hodge structure of H1,R on B0. Proof According to Matsushita’s Proposition 2.3, the restriction map H 2 (X, Q) → H 2 (Xb , Q) has rank 1. By Deligne’s invariant cycle theorem, this implies that the local system R 2 φ∗ R on B 0 has only one global section (provided by the polarization ω and its real multiples). If the local system (R 1 φ∗ R)∗ contains a nontrivial local subsystem L which carries a real variation of Hodge structures, the restriction of ω to L is nondegenerate and the orthogonal complement L⊥ω with respect to ω is also a local subsystem of (R 1 φ∗ R)∗ on which ω is nondegenerate. Thus we have a decomposition (R 1 φ∗ R)∗ = L ⊕ L⊥

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and if both1L and L⊥ were nontrivial, we would get two independent sections of R 2 φ∗ R = 2 ((R 1 φ∗ R)∗ )∗ , namely πL∗ ω|L and πL∗ ⊥ (ω|L⊥ ), where πL and πL⊥ are the two projections associated with the decomposition (23). This is a contradiction, hence a proper nontrivial such L does not exist. Let X be a very general hyper-Kähler manifold with Lagrangian fibration π : X → B and M ∈ Pic X a divisor which is topologically trivial on fibers of π. Assume π is not locally isotrivial. Let us prove that the assumptions of Theorem 1.6 are satisfied. If νM is not of torsion, the Zariski closure of ZνM (B) has an irreducible component which is a subfamily of abelian varieties (of positive dimension) of Pic0 (Xreg /Breg ). Such a family must be equal to Pic0 (Xreg /Breg ) by Lemma 5.5. This establishes assumption (ii) of Theorem 1.6. By Lemma 5.5, Pic0 (Xreg /Breg )

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has no locally trivial subfamily, because we are in the nonisotrivial case, so such a subfamily should be proper. This establishes assumption (i) of Theorem 1.6. Proof (Proof of Theorem 1.3 When dim X ≤ 8 in the Case of Maximal Variation) The maximal variation assumption tells us that at a general point b ∈ Breg the map dP : TB,b → Hom(H1,0,b , H0,1,b ) is injective. Reasoning by contradiction, if νM has the property that its torsion points are not dense in Breg , then as explained above, Theorem 1.6 applies and tells that the map ∇ : H1,0,b → Hom(TB,b , H0,1,b ) associated to dP above has the property that ∇ λ ∈ Hom(TB,b , H0,1,b ) is not an isomorphism for any λ ∈ H1,0,b . According to Theorem 5.4, these two maps are in fact identical and the ∇ λ ’s are the partial derivatives ∂λ g (3) of a cubic form g (3) on TB,b = H1,0,b . The homogeneous polynomials of degree 3 in n variables satisfying the condition that all of their partial derivatives are degenerate quadrics have been classified by Hesse and GordanNoether if n ≤ 4, and by Lossen [21] if n ≤ 5. We refer to section 7 for the latter. In the case where n ≤ 4, one has the following Theorem 5.6 (see [21]) Any cubic homogeneous polynomial in n ≤ 4 variables all of whose partial derivatives are degenerate is a cone. Being a cone means that one partial derivative is identically 0, which contradicts the injectivity of the map TB,b → Hom(H1,0,b , H0,1,b ). Remark 5.7 Part of the arguments described in this section also appear in Lin’s thesis [19].

6 Proof of Theorem 1.3 When the Fibration is Isotrivial 6.1 Further General Facts About Sections of Lagrangian Fibrations Let φ : X → B be a Lagrangian fibration, where X is a projective irreducible hyper-Kähler manifold of dimension 2n, and let B 0 ⊂ B be the Zariski open set of regular values of φ, X0 := φ −1 (B 0 ) ⊂ X. Denote by σ the holomorphic 2-form on X and by ω the first Chern class of an ample line bundle H on X. Over B 0 , the fibration φ : X0 → B 0 is a fibration into abelian varieties or rather a torsor on the corresponding Albanese fibration Alb(X0 /B 0 ), which admits a multisection Z → B 0 of degree d. Using these data, X0 → B 0 is isogenous to the associated Albanese fibration Alb(X0 /B 0 ) via the map iZ : X0 → Alb(X0 /B 0 )

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defined by iZ (u) = albXb (d{u} − Zb ), b = φ(u). Here the notation {u} is used to denote u as a 0-cycle in the fiber Xb (in order to avoid the confusion between addition of cycles and addition of points), so d{u} − Zb should be thought as a 0cycle of degree 0 on Xb , giving rise to an element albXb (d{u} − Zb ) ∈ Alb(Xb ). Next, using the polarisation H , the abelian fibration Alb(X0 /B 0 ) and its dual fibration Pic0 (X0 /B 0 ) are isogenous, the isogeny iH : Alb(X0 /B 0 ) → Pic0 (X0 /B 0 )

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being given by iH (b, vb ) = tv∗b (H|Xb ) − H|Xb for any b ∈ B 0 , vb ∈ Alb(Xb ). Here the translation tvb acts on Xb . The cycle H n−1 ∈ CHn−1 (X) also gives an isogeny albX0 /B 0 ◦ H n−1 : Pic0 (X0 /B 0 ) → Alb(X0 /B 0 )

(25)

which is easily shown to provides an inverse of (24) up to a nonzero coefficient. Note now that the (2, 0)-form σ on X is the pull-back of a (2, 0)-form σA on Alb(X0 /B 0 ) (or on Pic0 (X0 /B 0 )) via the rational map iZ : X  Alb(X0 /B 0 ). This indeed follows easily from Mumford’s theorem [27], since for any two points x, y of X0 such that iZ (x) = iZ (y) the difference albXb ({x} − {y}) is a torsion point in Alb (Xb ), b = φ(x) = φ(y), hence the cycle {x} − {y} is a torsion cycle in CH0 (Xb ) and a fortiori in CH0 (X) (in fact it actually vanishes in CH0 (X) since the later group has no torsion). Using the maps above, we get as well a nondegenerate (2, 0)-form σA on A := Pic0 (X0 /B 0 ), extending to a smooth projective completion, and for which the fibration is Lagrangian. Furthermore σ = (iH ◦ iZ )∗ σA . Our analysis of the properties of the holomorphic section ν of the torus fibration will be infinitesimal hence local, but we want first to add one restriction which comes from our global situation: The Lagrangian fibration φ : X → B being as above, let M ∈ Pic X be a line bundle on X which is topologically trivial on the fibers and let ν = νM : B 0 → A be the associated normal function. Lemma 6.1 The section ν is Lagrangian for the holomorphic 2-form σA .  be a desingularization of B. Then H 0 (B,  Ω 2 ) = 0 since a nonzero Proof Let B  B  would provide by pull-back via the rational map φ : holomorphic 2-form on B  a nonzero holomorphic form on X which is not proportional to σ . So X  B we just have to prove that ν ∗ (σA ) (which is a priori only defined as a holomorphic  However, we can provide 2-form on B 0 ) extends to a holomorphic 2-form on B. ∗  Looking another construction of ν (σA ) which will make clear that it extends to B: −1 at the definitions of iH , iZ , the subvariety Γ := (iH ◦ iZ ) (Im ν) is algebraic in  × X will provide a X0 and it is a multisection of φ over B 0 . Its Zariski closure in B

Torsion Points of Sections of Lagrangian Torus Fibrations and the Chow Ring. . .

317

 × X. Then as σ = i ∗ σA on X0 , we get that correspondence Γ ⊂ B Z ∗

Γ σ = Nν ∗ (σA )

(26)

on B 0 , where N is the degree of iH ◦ iZ . This gives the desired extension of ν ∗ (σA ).

6.2 Degenerate Normal Function and Degenerate Monge-Ampère Equation We complete the Donagi-Markman analysis in Sect. 5.1 by adding to the data described there the normal function ν : U → A, which satisfies the property that ν ∗ σA = 0, which is the global restriction coming from Lemma 6.1. This normal function ν ∈ Γ (U, A ) = Γ (U, H0,1 /H1,Z ) lifts locally to a section ν˜ of H0,1 ∼ = ΩU , hence provides locally a holomorphic 1-form α on U . Lemma 6.2 The form α is closed. Proof It is a general fact of the theory of integrable systems (see [14]) that the isomorphism (20) is also constructed in such a way that the pull-back σ˜ A of the holomorphic 2-form σA to the total space of the bundle H0,1 identifies with the canonical 2-form σcan on the cotangent bundle ΩU , which has the property that for any holomorphic 1-form β on U , seen as a section of ΩU , one has dβ = β ∗ (σcan ).

(27)

As the section ν˜ corresponds via this isomorphism to the 1-form α, the assumption that ν ∗ σA = 0 says equivalently that 0 = ν˜ ∗ (σ˜ A ) = α ∗ (σcan ) = dα. The closed holomorphic 1-form α thus provides us on simply connected open sets U of U with a holomorphic function f such that df = α. Remark 6.3 From now on, the holomorphic 2-form on our Lagrangian fibration will disappear. It is hidden in the fact that we work with the cotangent bundle of the base, which has a canonical holomorphic 2-form. We are in the local setting where the holomorphic functions fi , f constructed in the previous section exist. Let thus V be a complex manifold, and let f1 , . . . , f2n be holomorphic functions on V such that the functions xi := Re fi provide real coordinates on V , giving rise to a real variation of Hodge structures ev

R2n ⊗R OV → ΩV → 0, ei → dfi

(28)

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with underlying local system R2n = TV ,R and Hodge bundles H1,0 = Ker ev, H0,1 = ΩV . Let f be a holomorphic function on V ; as the dfi ’s are independent over R in ΩV ,b at any point b ∈ V , we can write the equality df =



ai dfi ,

(29)

i

of C ω forms of type (1, 0) on V 0 , where the ai ’s are real C ω functions on V , uniquely determined by (29). Denoting xi := Re fi , the xi ’s thus provide locally real coordinates on V 0 . Note that (29) is obviously equivalent, since the ai ’s are real and the considered forms are of type (1, 0), to the equality of real 1-forms: d(Re f ) =



ai dxi .

(30)

i

We assume that the map a· = (a1 , . . . , a2n ) : V 0 → R2n is of constant rank 2n − 2k (the rank is even by Lemma 3.2). The condition that a· is not of maximal rank at the general point is a degenerate real Monge-Ampère equation (see [28]). Lemma 6.4 The fibers of a· are affine in the coordinates xi . Proof As the map a· has locally constant rank near b ∈ V 0 , it factors through a submersion V 0 → Σ and an immersion Σ → R2n , where Σ is real manifold of dimension 2n − 2k. Equation (30) # says equivalently that for fixed σ = (λ1 , . . . , λ2n ) ∈ Σ, the function Re f − i λi xi has vanishing differential # on V 0 at any point of the fiber Vσ0 = a·−1 (λ1 , . . . , λ2n ). The function Re f − i ai xi on V 0 is thus the pull-back a·∗ ψ of a function ψ on Σ, that is Re f −



ai xi − a·∗ ψ = 0

(31)

i

on V 0 . If u = (u1 , . . . , u2n ) ∈ TΣ,σ , we get for any b ∈ Vσ0 , by differentiating (31) along any v ∈ TV 0 ,b such that a·∗ (v) = u and applying (30) 0=−



ui xi − dψ(u),

i

# which says that the function i ui xi is constant along the fiber Vσ0 . This provides 2n−2k real equations, affine in the xi ’s, vanishing along Vσ0 and independent over R, and they must define Vσ0 in V 0 since Vσ0 is a real submanifold of V of codimension 2n − 2k. The following lemma was partially proved in the proof of Lemma 3.2. 0 0 Lemma 6.5 # The fiber Vσ is a complex submanifold of V , along which the functions i ui fi are constant, for any u· ∈ TΣ,σ .

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# Proof The fact that the functions i ui fi are constant along Vσ0 for any u· ∈ TΣ,σ is proved exactly as the previous lemma, replacing (30) by (29). This implies that Vσ0 is a complex submanifold of V 0 since the real parts of these functions define Vσ0 as a real submanifold of V 0 by Lemma 6.4. Theorem 1.3 when the fibration is isotrivial works in any dimension. It will be obtained as a consequence of the following Proposition 6.6. Let φ : A → B be a locally isotrivial Lagrangian torus fibration, where A, B are quasiprojective and A has a generically nondegenerate closed holomorphic 2-form σA extending to a projective completion A and generating H 2,0 (A). Let ν : B → A be an algebraic section such that ν ∗ σA = 0. Proposition 6.6 If the torsion points of ν are not dense in B 0 for the usual topology, either the section ν is locally constant, or the local system H1,R,A := (R 1 φ∗ R)∗ admits a proper nontrivial local subsystem L which underlies a real subvariation of Hodge structures. Furthermore, writing the real lift (see Sect. 2) of the section ν as a sum ν˜ R,L + ν˜ R,L⊥ , where L⊥ is the orthogonal variation of Hodge structures with respect to ω, then ν˜ R,L is a locally constant section of L. Proof The local data described in Sect. 5.1 take the following form: there are locally 2n holomorphic functions f1 , . . . , f2n on simply connected open subsets U ⊂ B, which are independent over R and provide real coordinates x1 , . . . , x2n , xi = Re fi . By isotriviality, modulo the constants, the fi ’s provide only n independent holomorphic functions independent over C. In particular the open set B 0 over which φ is smooth and σA is nondegenerate is endowed with a flat holomorphic structure. Lemma 6.7 This flat holomorphic structure, after passing to a generically finite cover B of B 0 , is induced by the choice of n holomorphic 1-forms on a smooth projective completion B of B . Proof Indeed, the fibration φ : A → B is trivialized over a finite cover B of a Zariski open set of B 0 as A := A ×B B ∼ = B × J0 , where 0 ∈ B is a given point, and J0 is the fiber of φ : A → B over it. The local flat holomorphic coordinates on B 0 , pulled back to B , come from the holomorphic 2-form on B × J0 which is of the form 

prB∗ αi ⊗ prJ∗0 βi ,

i

for a basis βi of H 0 (ΩJ0 ). The flat structure on B 0 is given by the holomorphic forms αi . We now claim that the forms αi extend to holomorphic 1-forms on

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B . This is seen as follows: As the fibration is Lagrangian and σA generates H 2,0(A), the transcendental cohomology H 2 (A, Q)t r , pulled-back to B × J0 , falls in the (1, 1)-Künneth component H 1 (B , Q) ⊗ H 1 (J0 , Q) of the Künneth decomposition of H 2 (B ×J0 , Q), and this implies that it is contained in the image of H 1 (B , Q) ⊗ H 1 (J0 , Q) in H 1 (B , Q) ⊗ H 1 (J0 , Q) by Deligne’s strictness theorem for morphisms of mixed Hodge structures [13]): indeed, we get from the above a morphism of mixed Hodge structures α : H 2 (A, Q)t r ⊗ H 1 (J0 , Q)∗ → H 1 (B , Q) with image K ⊂ H 1 (B , Q) such that H 2 (A, Q)t r , pulled-back to B × J0 is contained in K ⊗ H 1 (J0 , Q). By Deligne strictness theorem [13], this morphism α has its image contained in the pure (smallest weight) part of H 1 (B , Q), namely H 1 (B , Q). This finishes the proof. As a consequence, we get the following: Corollary 6.8 Let (Ft )t ∈Σ be a continuous family of complex submanifolds of an open set V 0 of B 0 . Assume that each Ft is flat for the flat holomorphic structure described above, and is algebraic, that is, a connected component of the intersection of an algebraic subvariety Ct of B with V 0 . Then the Ft ’s are parallel, that is, recalling that the Ft are affine in the flat coordinates (see Lemma 6.4), the linear part of their defining equations is constant. Proof The linear part of the defining equations for Ft is determined, using Lemma 6.7, by the vector space of holomorphic forms on B vanishing on Ft , or equivalently on Ct . On the other hand, by desingularization and up to shrinking our family, we can assume that the Ct form a family of smooth projective varieties mapping to B . It is then clear that the space of holomorphic forms on B vanishing on Ct does not depend on t. The proof of Proposition 6.6 will finally use the following Lemma 6.9. Consider our locally isotrivial fibration φ : A → B and its section ν, which satisfies the condition ν˜ ∗ σA = 0. This section lifts locally on B 0 to a section ν˜M of the sheaf H1,R ⊗ CR∞ where H1,R is the the real local system (R 1 φ∗ R)∗ . This section is well-defined up to translation by a constant (integral) section of H1,R , and by Proposition 3.1, if the torsion points of νM are not dense in B 0 for the usual topology, then the map (defined on simply connected open sets V ⊂ B 0 ) ν˜ M : V → H1 (J0 , R) via a trivialization of H1,R has positive dimensional fibers Ft . Lemma 6.9 In the isotrivial case, the fibers Ft are algebraic.

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321

Proof Indeed, after passing to the finite cover B already introduced above, the normal function νM gives a morphism of algebraic varieties

: B → Pic0 (J0 ) ∼ νM =isog J0 ,

which to b ∈ B associates M|X with Xb ∼ = J0 canonically. It is immediate that b

, via the morphism B → B 0 and the fibers of ν˜ M coincide with the fibers of νM after restricting to the adequate open sets.

We can now apply Corollary 6.8 to the fibers Vσ0 ⊂ V 0 of the local map a· restricted to the open subset V 0 ⊂ V where a· has constant rank. These fibers are both complex and affine by Lemmas 6.4 and 6.5. We thus conclude that they vary in a parallel way. The tangent bundle to the fibers is thus a local subsystem of H1,R , and as the fibers are complex submanifolds, this local system is a subvariation of Hodge structure. This subvariation of Hodge structures is nontrivial by assumption. Saying that it is equal to H1,R is equivalent to saying that ν is locally constant. This finishes the proof. Proof (Proof of Theorem 1.3 in the Isotrivial Case) Let φ : X → B be a Lagrangian isotrivial fibration, X projective hyper-Kähler, and let νM be the normal function associated to a line bundle M on X which is topologically trivial on the smooth fibers of φ. Proposition 6.6 applies and shows that if the set of points b ∈ B where νM (b) is a torsion point is not dense in B, then either the normal function is locally constant or there is a proper nontrivial real subvariation of Hodge structure L of H1,R. We now use Lemma 5.5 which says that the second case is excluded and thus ν˜ is locally constant, so that νM is a section of R 1 φ∗ R/R 1 φ∗ Z. But then Lemma 3.3 says that a nontorsion section R 1 φ∗ R/R 1 φ∗ Z can exist only if the invariant part of H 1 (Xb , R) is not trivial, which provides a contradiction by Deligne’s global invariant cycles theorem since H 1 (X, R) = 0.

7 The Case of Dimension 10: Proof of Proposition 3.8 We prove in this section Proposition 3.8. We start with the proof of (i). We consider a projective hyper-Kähler tenfold X admitting a Lagrangian fibration π : X → B with maximal variation. Let M ∈ Pic X be cohomologous to 0 on the fibers Xb . We assume that the associated normal function νM : B 0 → Pic X0 /B 0 does not have a dense set of torsion points in B 0 . We use the induced Lagrangian fibration structure on the dual fibration Pic (X0 /B 0 ) → B 0 discussed in Sect. 6.1, that we denote by A → B 0 . We perform the same analysis as in Sect. 5. By Theorem 5.4, there are locally on B 0 holomorphic coordinates z1 , . . . , z5 and a holomorphic function (a ∂g potential) g with the following properties: Let gi = ∂z . Then the holomorphic i forms dz1 , . . . , dz5 , dg1 , . . . , dg5 are pointwise independent over R and the real

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variation of Hodge structure on H1,A is given by the evaluation map ev : R10 ⊗ OB → ΩB , ei → dzi , ei+5 → dgi , i = 1, . . . , 5. Thus the infinitesimal variation of Hodge structure TB → Hom (H1,0 , H0,1 ) ∗ ∼ H , by the homogeneous is given, using the natural identifications TB ∼ = H0,1 = 1,0

cubic form g (3) on TB determined at each point by the partial derivatives

∂3g ∂zi ∂zj ∂zk .

(3)

The maximal variation assumption says that the cubic form gb is not a cone at a general point b, while the non-density assumption for the torsion points implies by the André-Corvaja-Zannier Theorem 1.6 that it has the property that all of its partial derivatives are degenerate quadrics. For cubic forms in 5 variables, such cubics are classified in [21], which proves that such a cubic form on P4 has to be singular along a plane (and conversely, if a cubic in P4 is singular along a plane, then all of its partial derivatives vanish along a plane, hence are degenerate quadrics). We now prove the following result: Lemma 7.1 Let g(z1 , . . . , z5 ) be a holomorphic function defined on an open set U (3) of C5 with the property that at any point b ∈ U , the cubic form gb is not a cone and is singular along a projective plane Pb ⊂ P(TU,b ). Then there is locally on U a holomorphic map φ : U → C2 with fibers Sb affine in the coordinates zi , such ∂g that the partial derivatives ∂z are affine on each Sb . In particular, g restricted to i the fibers Sb is quadratic. Proof Let us consider the Taylor expansion of g in the coordinates zi centered at b ∈ B: g = qb + gb(3) + gb(4) + . . . ,

(32)

where qb is quadratic nonhomogeneous in the coordinates, and the homogeneous (k) forms gb in the variables Xi = zi − b depend polynomially on b for k ≥ 3. We observe that (3)

(4)

∂gb ∂g =± b . ∂zi ∂Xi

(33)

(3)

By assumption, the cubic form gb is singular along a projective plane Pb ⊂ P4 which is uniquely determined by b hence varies holomorphically with b (indeed, otherwise the cubic form gb(3) is a cone, as one checks easily). It follows that (4)

vanishes on Pb for all i, which means by (33) that the cubic for all i. Hence the quartic

gb(4)

∂gb ∂Xi

(3)

∂gb ∂zi

vanishes on Pb

is also singular along Pb . We can then continue and

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323

(k)

show inductively that all gb for k ≥ 3 are singular along Pb . In conclusion, fixing b, we have by (32) g = qb + gb ,

(34)

where qb is quadratic and gb vanishes doubly along the affine 3-space Sb := C3 ∩ U ⊂ U corresponding to Pb . This is saying that the map b → Pb is constant along Sb , which provides us with a foliation on U with affine fibers Sb , with the property that for each b, (34) holds along Sb with qb constant along Sb . Differentiating (34) ∂g ∂g b along Sb gives ∂z = ∂q ∂zi along Sb , which proves that ∂zi is affine on Sb . i Lemma 7.1 tells us that for our data (X, B, d) as above, B 0 (or rather the open set where the cubic is not a cone) is foliated by the locally defined fibers Sb , which clearly are both complex submanifolds and affine submanifolds of ∂g B 0 for the real affine structure. Indeed, the partial derivatives gi := ∂z are i 3 10 affine along Sb , so that each Sb maps to an affine C ⊂ C via the functions z1 , . . . , z5 , g1 , . . . , g5 , hence to a real affine subspace R6 ⊂ R10 via the real coordinates Re z1 , . . . , Re z5 , Re g1 , . . . , Re g5 . The proof of Proposition 3.8 (i) concludes with the following : Corollary 7.2 Along the leaves Fb of the foliation, the real variation of Hodge structure is a direct sum Lb,R ⊕ L b,R , where the real Hodge structure is locally constant on the first summand. Proof By Lemma 7.1, the leaf Fb , which identifies locally to Sb , locally maps, via the real affine structure given by (Re zi , Re gi ), 1 ≤ i ≤ 5, to an open set of an affine subspace R6 ⊂ R10 which will define L b,R ⊂ H1,R|Fb as the four dimensional subspace of real affine forms vanishing on Sb . The decomposition is then obtained using the polarization, Lb,R being defined as the orthogonal complement of L b,R . We have to prove that L b,R and Lb,R are real subvariations of Hodge structure of H1,R|Fb , and that the variation of Hodge structure on Lb,R is constant. The first point follows from the fact that Sb ⊂ U is also a complex submanifold of U so that the evaluation map L b,R ⊗R OB → ΩU maps to holomorphic 1-forms vanishing on Sb , which is a rank 2 vector subbundle of ΩU |Sb . The fact that the real variation of Hodge structure on Lb,R is constant is due to the fact that it identifies to the evaluation map Lb,R ⊗R OSb → ΩSb , ei → dfi , associated with a 6-dimensional real vector space of functions fi on Sb , where the fi ’s form only a 3-dimensional space of holomorphic functions on Sb since Sb maps via the fi ’s to an open subset of an affine C3 ⊂ C10 . Proof (Proof of Proposition 3.8, (ii)) We only have to prove that the normal function νM (or rather its lift ν˜ M,0,1 ∈ H0,1 ) decomposes locally along the leaves, h v v according to the decomposition of Corollary 7.2, as νM,0,1 + νM,0,1 , where νM,0,1 is constant. We use the local description introduced in Sect. 6.2: the lift ν˜ M,0,1 ∈ H0,1 ∼ = ΩB provides a holomorphic 1-form which is closed hence locally exact df . The variation of Hodge structure is given in local coordinates z1 , . . . , z5 by the

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evaluation map C10 ⊗ OB → ΩB , ei → dzi , e5+i → dgi , i = 1, . . . , 5, # ∂f ∂g i ∂zi dzi , the affine bundle ∂zi and g is as above. Writing df = # ∂f 10 H0,1,ν ⊂ C ⊗ OB introduced in (16) is the set of elements 5i=1 ∂z ei + i #5 #5 i=1 λi (ei+5 − j =1 gij ej ). We apply now Theorem 4.4 which tells us that the holomorphic map B × C5 → C10 where gi =

5 5 5    ∂f (b, λi ) → ei + λi (ei+5 − gij ej ) ∂zi i=1

(35)

j =1

i=1

is nowhere of maximal rank. Computing its differential at a point (b, λ· ), we conclude that the quadric fb(2) − ∂λ· (gb(3) ) (3)

is degenerate on TB,b for all λ· . As the cubic form gb at the general point b is not a cone but is singular along the plane Pb ⊂ P(TB,b ) defined by two homogeneous equations X1 , X2 , the cubic form has for equation X12 A+X22 B +X1 X2 C, where the linear forms A, B, C restrict to independent linear forms on Pb , hence the general (3) quadric ∂λ· (gb ) has rank 4 and its vertex is a general point of Pb . The condition that μfb(2) − ∂λ· (gb(3) ) is degenerate for any μ implies that the Hessian fb(2) of f at b vanishes on the vertex of the quadric defined by ∂λ· (gb(3) ), and as this is true for general λ· , fb(2) has to vanish on Pb . Thus the restricted function f|Sb has trivial Hessian, that is, f is affine on Sb . Thus d(f|Sb ) is a constant 1-form for the affine structure given by the zi ’s, which concludes the proof.

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