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Tannakian Categories of Perverse Sheaves On INAUGURAL-DISSERTATION zur Erlangung der Doktorwurde¨ an der Naturwissenschaftlich-Mathematischen Gesamtfakultat¨ der Ruprecht-Karls-Universitat¨ Heidelberg TANNAKIAN CATEGORIES OF PERVERSE SHEAVES ON ABELIAN VARIETIES vorgelegt von Dipl.-Math. Thomas Kramer¨ aus Frankenberg (Eder) Tag der mundlichen¨ Prufung:¨ Betreuer: Prof. Dr. Rainer Weissauer Zusammenfassung. Die vorliegende Dissertation beschaftigt¨ sich mit Tannaka-Kategorien, welche durch Faltung perverser Garben auf abelschen Varietaten¨ entstehen. Die Konstruktion dieser Kategorien ist eng verbunden mit einem kohomologischen Verschwindungssatz, der ein Analogon zum Satz von Artin-Grothendieck darstellt und als Spezialfall die generischen Verschwindungssatze¨ von Green und Lazarsfeld enthalt.¨ Die erhaltenen Tannaka-Gruppen bilden interessante geometrische Invarianten, welche in vielen Situationen eine Rolle spielen — nachdem die Grundlagen fur¨ das Studium dieser Gruppen bereitgestellt sind, wird als ein wichtiges Beispiel der Thetadivisor einer prinzipal polarisierten komplexen abelschen Varietat¨ behandelt. Durch Entartungsargumente wird die Tannaka-Gruppe fur¨ eine generische abelsche Varietat¨ bestimmt, welche wenigstens in Dimension 4 eine neue Antwort auf das Schottky-Problem liefert. Das Faltungsquadrat des Thetadivisors in Dimension 4 hangt¨ eng zusammen mit einer Familie glatter Flachen¨ vom allgemeinen Typ, und ein Studium dieser Familie fuhrt¨ auf eine Variation von Hodge-Strukturen mit Monodromiegruppe W(E6), die mit der Prym-Abbildung verbunden ist. Die Dissertation schließt ab mit einer Rekursionsformel fur¨ den generischen Rang der durch Faltungen von Kurven entstehenden Brill-Noether-Garben auf Jacobischen Varietaten.¨ Abstract. We study Tannakian categories that arise from convolutions of perverse sheaves on abelian varieties. The construction of these categories is closely intertwined with a cohomological vanishing theorem which is an analog of the Artin-Grothendieck theorem and contains as a special case the generic vanishing theorems of Green and Lazarsfeld. The arising Tannaka groups form a powerful new tool applicable in many different geometric contexts — after providing the framework for the study of these groups, we consider as an important example the theta divisor of a complex principally polarized abelian variety. Using degeneration techniques we determine the associated Tannaka group for a generic abelian variety, and we show that in dimension 4 this yields a Tannakian solution to the Schottky problem. The convolution square of the theta divisor in dimension 4 is closely related to a family of surfaces of general type, and a detailed study of this family leads to a variation of Hodge structures with monodromy group W(E6) which is connected with the Prym map. To conclude the dissertation, we provide a recursive formula for the generic rank of Brill-Noether sheaves which arise from the convolution of curves on Jacobian varieties. Acknowledgements It is a pleasure for me to thank my advisor, Professor R. Weissauer, for sharing his beautiful ideas in many productive mathematical discussions, for his general open-mindedness in all occuring questions, for his careful reading of the manuscript and for various helpful suggestions concerning the exposition. I am also grateful to my roommates Johannes Schmidt, Thorsten Heidersdorf and Uwe Weselmann and to the other members of the Mathematical Institute at Heidelberg for the pleasant environment which they created during the past years. Furthermore I would like to thank several people outside Heidelberg who enlivened the atmosphere on conferences and expressed their interest in my work, among them Anders Lundman, Bashar Dudin, Christian Schnell, Dan Petersen, Dmitry Zakharov, Giovanni Mongardi, Katharina Heinrich, Nicola Tarasca and Paul Hacking. I am also indebted to the Studienstiftung des deutschen Volkes and to all members of its staff for their flexibility and for the financial and ideal support from which I profited a lot. Finally, my most special thanks go to my family and friends for their encouragement and patience during my work, but perhaps even more for the multifarious, pleasant and interesting distractions from it without which I would not have been able to think freely. Contents Introduction 1 Some commonly used notations 7 Chapter 1. Vanishing theorems on abelian varieties 9 1.1. The main result 9 1.2. A relative generic vanishing theorem 11 1.3. Kodaira-Nakano-type vanishing theorems 12 1.4. Negligible perverse sheaves 15 1.5. The spectrum of a perverse sheaf 17 Chapter 2. Proof of the vanishing theorem(s) 23 2.1. The setting 23 2.2. Character twists and convolution 26 2.3. An axiomatic framework 30 2.4. The Andre-Kahn´ quotient 32 2.5. Super Tannakian categories 35 2.6. Perverse multiplier 36 2.7. The main argument 38 Chapter 3. Perverse sheaves on semiabelian varieties 41 3.1. Vanishing theorems revisited 42 3.2. Some properties of convolution 46 3.3. The thick subcategory of negligible objects 47 3.4. Tannakian categories 51 3.5. Nearby cycles 54 3.6. Specialization of characters 58 3.7. Variation of Tannaka groups in families 61 Chapter 4. The Tannaka group of the theta divisor 69 4.1. The Schottky problem in genus 4 71 4.2. Nondegenerate bilinear forms 73 4.3. Local monodromy 74 4.4. Degenerations of abelian varieties 79 4.5. Proof of the main theorem 85 4.6. Singularities of type Ak 88 v vi Contents Chapter 5. A family of surfaces with monodromy W(E6) 93 5.1. Intersections of theta divisors 95 5.2. Variations of Hodge structures 102 5.3. Integral cohomology 106 5.4. Neron-Severi´ lattices 109 5.5. The upper bound W(E6) 115 5.6. Negligible constituents 119 5.7. Jacobian varieties 123 5.8. The difference morphism 127 5.9. A smooth global family 131 Chapter 6. The generic rank of Brill-Noether sheaves 135 6.1. Brill-Noether sheaves 135 6.2. A formula for the generic rank 137 6.3. Computations in the symmetric product 139 6.4. The fibres of multiple Abel-Jacobi maps 142 6.5. Betti polynomials 144 6.6. Some numerical values 146 Appendix A. Reductive super groups 151 Appendix B. Irreducible subgroups with bounded determinant 157 Appendix C. An irreducible subgroup of W(E6) 163 Appendix D. Doubly transitive permutation groups 165 Bibliography 169 Introduction The geometry of a smooth complex projective curve C with Jacobian variety Jac(C) is governed, after the choice of a base point on the curve, by the Abel-Jacobi morphism. In particular, the latter defines for each n 2 N a morphism Cn = C × ··· ×C −! Jac(C) whose fibres correspond to linear series of divisors on the curve, and the loci where the fibre dimension jumps r are the subvarieties Wn ⊆ Jac(C) of special divisors that have been studied extensively in classical Brill-Noether theory [6]. For smooth complex projective varieties Y of higher dimension one still has the Albanese morphism n fn : Y = Y × ··· ×Y −! X = Alb(Y); but here the situation is much harder to describe in general. For example, the image f1(Y) ,! X can have any dimension between zero and dim(X), and there is no obvious substitute for Brill-Noether theory in general. In some sense, the first three chapters of this thesis provide a possible framework for such a substitute. More specifically, the properties of fn we are interested in (such as the loci on X where the fibre dimension jumps) can be studied via the higher direct images i R fn∗(CY n ) for i ≥ 0; which are constructible sheaves in the sense that there exists a stratification of X into finitely many locally closed subvarieties over which they restrict to locally constant sheaves. In general the geometry of these higher direct image sheaves is rather involved, as we will see in a particular example in chapter 5. However, we propose a simple description of such direct images in terms of the representation theory of a reductive algebraic group that can be attached to Y via the Tannakian formalism [33]. The basis for this Tannakian description will be a vanishing theorem for constructible sheaves on abelian varieties which can be seen as an analog of the Artin-Grothendieck affine vanishing theorem [7, exp. XIV, cor. 3.2] and contains as a special case the generic vanishing theorems of Green and ∗ Lazarsfeld [47]. For any character c : p1(X;0) −! C of the fundamental group, let us denote by Lc the corresponding local system on X. Then in chapters 1 and 2 which are based on joint work with R. Weissauer [68], 1 2 Introduction we show that for any constructible sheaf F on X and a sufficiently general character c one has i H (X;F ⊗C Lc ) = 0 for all i > dim(Supp(F)): An independent proof of this result has also been given by C. Schnell [94], using the Fourier-Mukai transform for holonomic D-modules. Our proof is of a different flavour and is closely intertwined with the definition of the Tannakian categories we are interested in. It is based on an abstract quotient construction for semisimple tensor categories — we only use D-modules at a single place in section 1.4 where we classify all perverse sheaves of Euler characteristic zero (which will be precisely the objects which become isomorphic to zero under the above quotient construction). At present we do not know whether this classification extends to algebraically closed base fields of positive characteristic p > 0, but otherwise all our arguments also work for l-adic constructible sheaves on abelian varieties over the algebraic closure of a finite field Fp for prime numbers l 6= p. Our Tannakian results are best formulated in the framework of perverse sheaves which has its historic roots in the theory of D-modules [57] and in the sheaf-theoretic construction of intersection cohomology for singular varieties [10]. Let us for convenience briefly recall some basic definitions and notations from loc. cit. For any complex algebraic variety Z, we will b denote by Dc (Z;C) the derived category of bounded C-sheaf complexes whose cohomology sheaves are constructible for some stratification of Z into Zariski-locally closed subsets.
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