Topics Related to Vector Bundles on Abelian Varieties

TOPICS RELATED TO VECTOR BUNDLES ON ABELIAN VARIETIES

NATHAN MARK GRIEVE

A thesis submitted to the

Department of Mathematics and Statistics in conformity with the requirements for the degree of Doctor of Philosophy

Queen’s University

Kingston, Ontario, Canada June 2013

Abstract

This thesis is comprised of three logically independent parts. As the title suggests, each part is related to vector bundles on abelian varieties. We ﬁrst use Brill-Noether theory to study the geometry of a general curve in its canonical embedding. We prove that there is no g for which the canonical embedding of a general curve of genus g lies on the Segre embedding of any product of three or more projective spaces. We then consider non-degenerate line bundles on abelian varieties. Central to our work is Mumford’s index theorem. We give an interpretation of this theorem, and then prove that non-degenerate line bundles, with nonzero index, exhibit positivity analogous to ample line bundles. As an application, we determine the asymptotic behaviour of families of cup-product maps. Using this result, we prove that vector bundles, which are associated to these families, are asymptotically globally generated. To illustrate our results, we consider explicit examples. We also prove that simple abelian varieties, for which our results apply in all possible instances, exist. This is achieved by considering a particular class of abelian varieties with real multiplication.

The ﬁnal part of this thesis concerns the theory of theta and adelic theta groups. We extend and reﬁne work of Mumford, Umemura, and Mukai. For example, we i determine the structure and representation theory of theta groups associated to a class of vector bundles which we call simple semi-homogeneous vector bundles of separable type. We also construct, and clarify functorial properties enjoyed by, adelic theta groups associated to line bundles.

ii Acknowledgments

First and foremost I thank my PhD advisor Mike Roth for his supervision, teaching, encouragement, and also for many helpful discussions regarding this work. Conversations with Greg Smith, Ernst Kani, and A.T. Huckleberry also con-

tributed to this thesis. It is my pleasure to thank Leslie Roberts for his supervision while I was his Master’s student. I have also learned a great deal from Ivan Dimitrov and Tony Geramita.

The Department of Math and Stats at Queen’s University is a delightful place to learn, conduct research, and teach. I have benefited immensely from teaching two courses and also from attending and giving research talks at the Algebraic Geome- try Seminar and the Number Theory Seminar. Of course talking to other graduate students, including Chris Dionne, Hector Pasten, Ilia Smirnov, and Andrew Staal, has also influenced this work. I am especially fortunate to have met Valdemar Tsanov. I have benefited both personally and mathematically from our many interactions. I have been supported financially by an E.G. Bowman Fellowship (2008-2009) and several Ontario Graduate Fellowships (2009-2013). Finally I thank my family for their support and encouragement. I especially iii thank my wife Stacy Grieve for her love and inspiration.

iv Statement of Originality

I hereby certify that all of the work described within this thesis is the original work of the author. Any published (or unpublished) ideas and/or techniques from the work of others are fully acknowledged in accordance with the standard referencing practices.

v Table of Contents

Abstract i

Acknowledgments iii

Statement of Originality v

Table of Contents vi

1: Introduction ...... 1

1.1 Historicalsketch...... 2 1.1.1 ThePetri-map: specialdivisorsoncurves ...... 2

1.1.2 Abelian varieties: cup-products arising from ample line bundles...... 3 1.2 Theproblemsaddressedinthisthesis ...... 6 Guidingproblems...... 6 Summaryanddiscussionofresults ...... 6

1.2.1 Curves: Segre embeddingsandcanonicalcurves ...... 6 1.2.2 Abelian varieties: cup-products arising from non-degenerate linebundles ...... 7

vi 1.2.3 Abelian varieties: theta and adelic theta groups ...... 10

2: Background ...... 12

2.1 Preliminaryfactsaboutlinebundlesoncurves ...... 12

2.2 Brill-Noethertheory...... 13 2.3 Preliminary facts about line bundles on complex tori ...... 15 2.4 Preliminary facts about line bundles on abelian varieties ...... 20 2.5 Preliminary facts about vector bundles on abelian varieties ...... 21

2.6 Othernotationandconventions ...... 22

Curves: Segreembeddingsandcanonicalcurves ...... 24

3.1 Statementofresultsandoutlineoftheirproof ...... 24 3.2 The n-fold Petri-map and proof of Proposition 3.1 ...... 25 3.3 ProofofTheorem3.2anditscorollary ...... 28

4: Abelian varieties: index conditions and cup-products ...... 32

4.1 Thepairindexcondition ...... 32 4.2 Statementofresultsandoutlineoftheirproof ...... 34 4.3 Mumford’s index theorem and the real Neron-Severi space ...... 36 4.3.1 Non-degenerate R-divisors ...... 37

1 4.3.2 The index of non-degenerate elements of N (X)Q ...... 38

1 4.3.3 The index function and the non-degenerate locus of N (X)R . 38 4.4 Familiesofcup-productproblems...... 40

vii 4.5 Proofoftheorems ...... 43 4.5.1 Twopreliminaryremarks ...... 43

4.5.2 ProofanddiscussionofTheorem4.2 ...... 45 4.5.3 ProofanddiscussionofTheorem4.1 ...... 48 4.5.4 ProofanddiscussionofTheorem4.3 ...... 50 4.5.5 Theorem4.3andtheworkofPareschi-Popa ...... 52

5: Examples of cup-product problems on abelian varieties ...... 55

5.1 Cup-productsarenot,ingeneral,nonzero ...... 55 5.1.1 The Neron-Severi space of X ...... 56 5.1.2 Cup-products on X ...... 56 5.1.3 Cup-products with nonzero source and target space can re-

sultinazeromap...... 57 5.1.4 TheﬁrstChernclass ...... 58 5.2 Thepairindexconditionandnumberﬁelds ...... 59 5.3 The classical case: theta groups and cup-products ...... 61

5.3.1 Preliminaries...... 61 5.3.2 Cup-products, ample line bundles, and theta groups ...... 62 5.3.3 Cup-products, theta groups, and line bundles with nonzero index ...... 62

6: Theta groups andvector bundleson abelianvarieties ...... 65

6.1 Formulationandstatementofresults ...... 65

viii 6.2 Constructionofthetagroups...... 67 6.2.1 Preliminariesfromdescenttheory ...... 67

6.2.2 Thetagroupsandquasi-coherentsheaves ...... 70 6.2.3 Levelsubgroupsanddescent ...... 70 6.2.4 Thetagroupsandisogenies ...... 71 6.3 Non-degeneratethetagroups ...... 73

6.3.1 Preliminaries on central extensions of k× by a ﬁnite abelian group...... 74 6.3.2 Non-degenerate theta groups and vector bundles ...... 77 6.4 Representations of non-degenerate theta groups ...... 79

6.4.1 AuxiliaryresultsandproofofTheorem6.11 ...... 80 6.4.2 Inducedrepresentations ...... 87 6.4.3 ProofofTheorem6.2andCorollary6.3 ...... 88

7: Adelicthetagroupsandlinebundles ...... 90

7.1 Formulationandstatementofresults ...... 90

7.2 Preliminaries...... 91 7.3 Diagrams of k×-torsors ...... 93 ∗ f ∗ 7.3.1 The pull-back morphisms Autf(x)(L) Autx(f L) ...... 94 L,x −→ ∗ an,m ∗ 7.3.2 The morphisms Autxm (mX L) Autxn (nX L) ...... 97 −−→f,L bx 7.3.3 The morphisms Aut (m∗ f ∗L) ,m Aut (m∗ L) ..... 99 xm X −−→ f(xm) Y 7.4 Adelicthetagroups ...... 101 7.4.1 The group G(L) ...... 102 7.4.2 The skew-symmetric bilinear form [ , ] ...... 107 b − − G(L) b ix 7.4.3 The group homomorphism G(f) ...... 107 The group homomorphism NS(X) ֒ H2(V(X); k×) ...... 108 7.5 b→

8: Topicsforfutureinvestigation ...... 112

8.1 More generalinstances ofthe pairindexcondition ...... 112 8.2 Embeddings of abelian varieties, pull-backs, and harmonicforms . . 114

Bibliography ...... 116

x 1

Introduction

Vector bundles on abelian varieties interact in interesting ways with many areas of mathematics including the theory of algebraic curves, number theory, and repre-

sentation theory. In this thesis, we see glimpses of these interactions in each topic that we consider. We ﬁrst address a problem which concerns the geometry of a general curve in its canonical embedding. Our solution to this problem is related, via Brill-Noether theory, to the geometry of certain moduli schemes which are determinantal subva- rieties of abelian varieties. The main focus of our second topic is a cohomological condition which we place on a pair of vector bundles on an abelian variety. This condition is motivated by its relation to cup-product maps. This relation is the subject of Chapter 4. On the other hand, as we see in Chapter 5, this condition is also related to number ﬁelds. Theta and adelic theta groups are the subject of our third topic. These groups are historically important in the study of moduli and syzygies of abelian varieties.

1 1. INTRODUCTION 2

In Chapters 6 and 7 we examine these groups in detail. It is in these chapters that the relationship between representation theory and vector bundles on abelian

varieties is most apparent. To place this thesis in proper context, we survey relevant parts of the literature in 1.1. We then state the main problems and describe some of our results in 1.2. § §

1.1 Historical sketch

The geometry of a projective variety X is often related to cup-product maps.

Problem 1.1. Let L and M be line bundles on X. Describe the image of the cup- product map ∪ H0(X, L) H0(X, M) H0(X, L M). ⊗ −→ ⊗ For instance is this map nonzero? Is it injective? Is it surjective?

We now discuss two particular cases of Problem 1.1. As it turned out, these cases motivated and inspired several results in this work.

1.1.1 The Petri-map: special divisors on curves

Let C be a smooth irreducible curve deﬁned over the complex numbers and let KC denote the canonical line bundle of C. Every line bundle L on C determines a cup-product map

∪ µ(L) : H0(C, L) H0(C, L∨ K ) H0(C,K ) (1.1) ⊗ ⊗ C −→ C which is known as the Petri-map. The source of the map (1.1) is zero unless

H0(C, L) =0 and H0(C, L∨ K ) =0. (1.2) 6 ⊗ C 6 1. INTRODUCTION 3

Line bundles L on C for which (1.2) holds are called special. If C is a general curve, then the map (1.1) was conjectured by Petri to be injective for all line bundles L on C. This conjecture was ﬁrst proved by Gieseker [15]. The map (1.1) has remarkable geometric implications especially in the setting of special line bundles. Special line bundles are the subject of Brill-Noether theory. We give a more detailed explanation of these topics in 2.1 and 2.2. § §

1.1.2 Abelian varieties: cup-products arising from ample line bun-

dles

Let X be an abelian variety and let L and M be ample line bundles on X. Cup- product maps of the form

∪ H0(X, L) H0(X, M) H0(X, L M) (1.3) ⊗ −→ ⊗ have been extensively studied because they are related to the syzygies of X in a suitable projective embedding. We now describe some parts of this story because it is related to what we do here. (See 5.3 below as well.) § The starting point is work of D. Mumford [36], [37], [38], [39]. In these papers he systematically studied maps of the form (1.3) as part of proving a number of results related to moduli of abelian varieties, syzygies of abelian varieties, and theta functions. Three important features to Mumford’s approach were: the theta group of a line bundle; the adelic theta group of a polarized tower of abelian varieties; the observation that cup-product maps of the form (1.3) can be studied by considering families of cup-product maps parametrized by the dual abelian variety. Theta and adelic theta groups are interesting objects in their own right and 1. INTRODUCTION 4

were constructed by Mumford [36, 1], [39, p. 64], [37, 7], [41, 23], [40, Chap- § § § ter 4]. H. Umemura extended Mumford’s theory of theta groups by constructing theta groups associated to vector bundles on X. He determined the weight 1 representation theory of theta groups associated to simple vector bundles on X. Additionally, he posed the problem of determining the structure of theta groups associated to vector bundles in general and considered these groups in the context of Brauer-Severi varieties over abelian varieties [55, 5], [56, 1 and 2]. More re- § § § cently, M. Brion considered theta groups associated to Brauer-Severi varieties over abelian varieties [9]. S. Shin also has used ideas of Mumford to construct adelic theta groups and adelic-Weil-representations associated to abelian schemes [52].

Returning to Mumford’s work concerning syzygies of abelian varieties, let us point out that his work led to a number of results and successful generalizations. Notable examples include work of S. Koizumi [24], G. Kempf [21], Lazarsfeld- Pareschi [44], and Pareschi-Popa [45], [46]. These generalizations were related to questions which can be phrased in terms of the property Np of M. Green and R. Lazarsfeld [16], [28, p. 116]. For example, Lazarsfeld conjectured that if A is an ample line bundle on a

ℓ complex abelian variety X then the line bundle A satisﬁes property Np whenever ℓ > p+3, [27, Conjecture 1.5.1, p. 516]. This conjecture was proven by Pareschi [44] and then latter improved by Pareschi-Popa: using the fact that Aℓ satisﬁes property

ℓ Np when ℓ > p +3 they proved that A satisﬁes property Np when ℓ = p +2, [46, Theorem 6.2, p. 184]. Central to their proof is the study of cup-product maps of the form

∪ H0(X, An) H0(X, E) H0(X, An E) (1.4) ⊗ −→ ⊗ 1. INTRODUCTION 5

where A is a globally generated ample line bundle on X and where E is a suitably deﬁned vector bundle on X.

To study these maps, Pareschi-Popa formulated index conditions, for instance the IT, WIT, and PIT [44, p. 653] and [46, p. 179]. (Compare also with work of S. Mukai [34, p. 156].) They used these conditions to study families of cup-product maps of the form

∪ H0(X, T ∗(An)) H0(X, E) H0(X, T ∗(An) E) (1.5) x ⊗ −→ x ⊗ as x X varies. ∈ As an example, their index conditions allowed them to relate the cup-product map (1.5) to the ﬁberwise evaluation map of the vector bundle

p (m∗An p∗E) 1∗ ⊗ 2 on X; here p denote the projection of X X onto the ﬁrst and second factors i × respectively (for i = 1, 2) and m denotes multiplication in the group law. They then gave cohomological criteria for vector bundles on X to be globally generated. That cup-product maps of the form (1.5) were surjective then became a question of applying this criteria to the vector bundle p (m∗An p∗E). 1∗ ⊗ 2 Pareschi-Popa subsequently generalized their approach and went on to study families of cup-product maps having the form

H0(X, T ∗E) H0(X, F ) ∪ H0(X, T ∗(E) F ) x ⊗ −→ x ⊗ where E and F are semi-homogenous vector bundles on X and x X varies [47, ∈ 7.3]. § 1. INTRODUCTION 6

1.2 The problems addressed in this thesis

Guiding problems

Let X be a projective variety deﬁned over an algebraically closed ﬁeld k.

Problem 1.2. Let E and F be vector bundles on X, and let p, q N. Describe the ∈ image of the cup-product map

∪ (E, F ) : Hp(X, E) Hq(X, F ) Hp+q(X, E F ). ∪p,q ⊗ −→ ⊗

For instance is (E, F ) nonzero? Is it surjective? Is it injective? To what extent is ∪p,q (E, F ) related to the geometry of X? ∪p,q

To pose non-trivial cup-product problems, we must ensure that the source and

target space of such maps are nonzero.

Problem 1.3. For ﬁxed p, q N does X admit a pair of vector bundles (E, F ) for ∈ which (E, F ) has nonzero source and target space? ∪p,q

This thesis addresses topics related to Problems 1.2 and 1.3 in the particular case that X is a curve or an abelian variety.

Summary and discussion of results

1.2.1 Curves: Segre embeddings and canonical curves

In Chapter 3, we study the image of a general curve of genus g > 3 in its canonical embedding. Our main result concerns the geometry of these curves and relies on the main theorems of Brill-Noether theory (see 2.2). § 1. INTRODUCTION 7

To place our results into perspective, let us ﬁrst remark that if g is composite then the canonical embedding of a general curve of genus g lies on the Segre embedding of a product of two projective spaces. For example, a general curve of genus 4 lies on the Segre embedding of P1 P1 while a general curve of genus 6 × lies on the Segre embedding of P1 P2. These facts have applications concerning × the structure of the Chow ring of as illustrated in [12, p. 421] and [48, p. 26]. Mg The purpose of Chapter 3 is to prove that, by contrast, there is no g for which the canonical embedding of a general curve of genus g lies on the Segre embedding of any product of three or more projective spaces. This theorem is obtained as a consequence of a stronger result which describes the possible linear series which can arise from an n-tuple of line bundles on a general curve once we impose a restriction on the tensor product of this n-tuple. (See Theorem 3.2 and Corollary 3.3.) The original motivation for this project was to settle curiosities related to the n-fold Petri-map. Using the injectivity of the case n = 2 we prove that this map is injective for all n-tuples of line bundles on a general curve. (Lemma 3.4.) The results of Chapter 3 appear in [17].

1.2.2 Abelian varieties: cup-products arising from non-degenerate

line bundles

To describe what we do in Chapters 4 and 5, we first make some preliminary remarks. Let X be an abelian variety defined over an algebraically closed field k. A line bundle L on X is non-degenerate if χ(L) =0. By work of D. Mumford [41, 16], if L 6 § 1. INTRODUCTION 8

is a non-degenerate line bundle, then there exists a unique integer i(L) for which Hi(L)(X, L) =0. 6 In Chapters 4 and 5, we study questions related to cup-product problems arising from line bundles L and M on X satisfying the condition that

L, M, and L M are non-degenerate and i(L)+i(M)=i(L M). (1.6) ⊗ ⊗

This condition ensures that the cup-product map

∪ Hi(L)(X, L) Hi(M)(X, M) Hi(L⊗M)(X, L M) (1.7) ⊗ −→ ⊗ has nonzero source and target space.

The above discussion prompts

Problem 1.4. Let X be an abelian variety of dimension g. (a) For what integers p, q > 0, p + q 6 g, does X admit line bundles L and M such that L, M, and L M are non-degenerate, i(L) = p, i(M) = q, and ⊗ i(L M)= p + q? ⊗ (b) Let L and M be line bundles on X satisfying condition (1.6). Describe the image of the cup-product map (1.7). For instance, is it nonzero?

We address Problem 1.4 and generalizations thereof. We prove four theorems

all of which are related to Problem 1.4. (Theorems 4.1, 4.2, 4.3, and 5.1.) In fact, condition (1.6) is one instance of a more general condition which we call the pair index condition (PIC) (see 4.1). We now describe some of the results of Chapters 4 § and 5 in more detail. Theorem 4.1 addresses the asymptotic nature of the PIC. It implies that if E and

F are vector bundles on X and if (L, M) is a pair of line bundles which satisﬁes 1. INTRODUCTION 9

condition (1.6) then the cup-product maps

∪ Hi(L)(X, T ∗(Ln E)) Hi(M)(X, M n F ) Hi(L⊗M)(X, T ∗(Ln E) M n F ) (1.8) x ⊗ ⊗ ⊗ −→ x ⊗ ⊗ ⊗ have nonzero source and target space for all x X and all n 0. ∈ ≫ Theorem 4.3 concerns the behaviour of the cup-product map (1.8). It implies that the map (1.8) is nonzero and surjective for all x X and all n 0. ∈ ≫ Additionally, for n ﬁxed, we prove that the condition that the map (1.8) is sur-

jective for all x X is equivalent to the condition that the vector bundle ∈ Ri(L⊗M)(m∗(Ln E) p∗(M n F )) (1.9) p1∗ ⊗ ⊗ 2 ⊗ is globally generated (Proposition 4.8). Thus, Theorem 4.3 also implies that the vector bundle (1.9) is globally generated for all n 0. ≫ Both Theorem 4.1 and Theorem 4.3 rely heavily on Theorem 4.2. This theorem also implies that if L is a non-degenerate line bundle on X and i(L) 6 q then L is q-ample in the sense of [53]. We prove Theorems 4.1, 4.2, and 4.3 in 4.5. It is also important to note that § these theorems apply to line bundles with nonzero index as well as to abelian varieties deﬁned over algebraically closed ﬁelds of positive characteristic. One interpretation of these results is that non-degenerate line bundles with nonzero index exhibit positivity analogous to that of ample line bundles; this positivity follows from, and can be expressed in terms of, Mumford’s index theorem.

We consider several examples in Chapter 5. These examples illustrate Theo- rems 4.1, 4.2, and 4.3 as we now explain. When considering cup-product maps such as (1.7), arising from a pair of line bundles (L, M) satisfying condition (1.6) and having nonzero index, there is no reason to expect that such maps should be nonzero–in 5.1 we construct 1-parameter § 1. INTRODUCTION 10

families of such cup-product problems which result in a zero map. In 5.3, we discuss an approach, used by Mumford, to study cup-product prob- § lems arising from pairs of ample line bundles. We discuss the extent to which these techniques are applicable in our more general setting. This example is also related to what we have discussed in 1.1.2. § In 5.2, we address Problem 1.4 (a). In doing so, we relate condition (1.6) to § number ﬁelds. For example, a consequence of Theorem 5.1 is that, for g ﬁxed, there exists complex simple abelian varieties, of dimension g and having real multiplication, for which every possible instance of Theorems 4.1, 4.2, and 4.3 can actually occur.

1.2.3 Abelian varieties: theta and adelic theta groups

The purpose of Chapters 6 and 7 is to refine and generalize the theory of theta and adelic theta groups. Specifically, we consider theta groups associated to a class of simple semi-homogeneous vector bundles which we refer to as simple semi- homogeneous vector bundles of separable type. We determine both the structure and representation theory of these groups. Our first result, Theorem 6.1, generalizes [36, Theorem 1]. It also answers a problem posed by Umemura [55, p. 120] for simple semi-homogeneous vector bundles of separable type. Our second result, Theorem 6.2 and Corollary 6.3, generalizes Proposition 3 and Theorem 2 of [36, 1]. It also makes the inequality as- § serted in [6, Exercise 6.10.4, p. 175] an equality. We also give a conceptual interpretation of these irreducible representations in terms of induced modules; they are induced by 1-dimensional representations of an appropriate subgroup. (See 1. INTRODUCTION 11

6.4.2.) To see an example of higher weight representations of such groups occur- § ing in nature, see [19, Example 2.10, p. 349]. It is important to note that our results build on work of Mukai [33] and Mumford [36, 1]. § We then revisit Mumford’s theory of adelic theta groups. In fact, we start by constructing such groups. Our construction is inspired by ideas of Mumford, see [37, 7] and [40, Chapter 4], and grew out of efforts to understand these works. § Our construction also differs from the more recent constructions of S. Shin [52]. We use our construction to prove Theorem 7.1. To describe this result, let I denote the set of positive integers which are not divisible by the characteristic of k. Let

tor(X) := x X(k) : nx =0 for some n I , { ∈ ∈ } and let V(X) := lim tor(X), where the limit is indexed by I and the maps are given

←− n by multiplication by m whenever m divides n. In is notation, Theorem 7.1 gives a functorial realization of the Neron-Severi group of X as a subgroup of H2(V(X); k×). 2

Background

In order to keep this thesis reasonably self-contained, we provide some relevant background material.

2.1 Preliminary facts about line bundles on curves

Let C be a smooth irreducible curve deﬁned over the complex numbers. If L is a

i i line bundle on C, then let h (C, L) := dimC H (C, L). Let KC denote the canonical

0 line bundle of C and let g := h (C,KC ) denote its genus. Let L be a line bundle on C. The Riemann-Roch formula asserts

χ(L)= h0(C, L) h0(C, L∨ K ) = deg L g +1. (2.1) − ⊗ C −

12 2. BACKGROUND 13

Using the formula (2.1), we deduce

if deg L> 2g 2, then h0(C, L) = deg L g +1 and h1(C, L)=0;  − −  if deg L< 0, then h0(C, L)=0 and h1(C, L)= deg L + g 1; (2.2)  − −  if 0 6 deg L 6 2g 2, then we need more information.  −   IfL has two nonzero cohomology groups, then we say that L is a special line bundle. Using (2.2) we deduce that, if L is a special line bundle, then

0 6 deg L 6 2g 2. −

Special line bundles are the subject of Brill-Noether theory.

2.2 Brill-Noether theory

We refer to [4, Chap. IV] and [3, Chap. XXI] for the basic results in Brill-Noether theory and [4, p. xv] for notation. Here we give a quick exposition of some of the main results of the theory.

r For all smooth curves C of genus g, there exist moduli schemes Wd (C) whose closed points are identiﬁed with the set

L : L Pic(C), deg L = d and h0(C, L) > r +1 . { ∈ }

r For the construction of Wd (C), see [14, p. 279] or [4, p. 176].

r Brill-Noether theory studies the geometry of the schemes Wd (C). Central to the theory is the Brill-Noether number and the Petri-map. The Brill-Noether number is deﬁned by ρ(g,r,d) := g (r + 1)(g + r d). See − − [4, p. 159] for an explanation as to how this number arises. On the other hand, the 2. BACKGROUND 14

Petri-map is deﬁned for all line bundles L on C; it is the cup-product map

µ(L) : H0(C, L) H0(C, L∨ K ) H0(C,K ). ⊗ ⊗ C → C

Both the Brill-Noether number and the Petri-map have remarkable geometric implications as seen in the two main theorems of Brill-Noether theory which we now describe. The ﬁrst theorem applies to all smooth curves and is the result of work by Kempf, Kleiman-Laksov, and Fulton-Lazarsfeld.

Theorem 2.1 (Brill-Noether theory theorem I [23], [14]). Let C be a smooth curve of genus g.

(a) If ρ(g,r,d) > 0, then W r(C) = ∅ and every irreducible component of W r(C) has d 6 d dimension greater than or equal to ρ(g,r,d).

r (b) If ρ(g,r,d) > 1, then Wd (C) is connected.

If C is a general curve, then the converse to Theorem 2.1 (a) holds–we state this in the second main theorem (Theorem 2.4). That the converse to Theorem

2.1 (a) holds for general curves was first proved by Griffiths-Harris [18] using a degeneration argument. Petri conjectured that µ(L) is injective for all line bundles L on a general curve C. (This is implicit in [49]. See the footnote on [4, p. 215] for a discussion.) Arbarello-Cornalba clarified the geometric implications of this conjecture.

Theorem 2.2 (Arbarello-Cornalba [2, Theorem 0.3]). Let C be a general curve of genus g. If µ(L) is injective for all line bundles L on C and W r(C) = ∅, then every irre- d 6 r r ducible component of Wd (C) has dimension ρ(g,r,d) and Wd (C) is non-singular away r+1 from Wd (C). 2. BACKGROUND 15

In [15], Gieseker used a degeneration argument, that was subsequently stream- lined by Eisenbud-Harris [11], to prove that µ(L) is injective for all line bundles

on a general curve. Lazarsfeld, without using degenerations, also gave an independent proof [26, p. 299]. The fact that µ(L) is injective for all line bundles on a general curve is sometimes referred to as the Gieseker-Petri theorem.

Theorem 2.3 (Gieseker-Petri [15, Theorem 1.1, p. 251]). If C is a general curve of genus g, then the cup-product µ(L) is injective for all line bundles L on C.

The above discussion, combined with [14, Corollary 2.4, p. 280], is summarized in the second main theorem of Brill-Noether theory.

Theorem 2.4 (Brill-Noether theory theorem II [2], [14], [15], [18], [26]). Let C be a general curve of genus g.

(a) If W r(C) = ∅, then ρ(g,r,d) > 0, W r(C) is of pure dimension ρ(g,r,d), and d 6 d r r+1 Wd (C) is non-singular away from Wd (C).

r (b) If ρ(g,r,d) > 1, then Wd (C) is irreducible.

2.3 Preliminary facts about line bundles on complex

tori

We refer to [41] and [6] for proofs of the material we describe in this section. A lattice Λ Cg = R2g is the Z-span of an R-basis for Cg.A complex torus is a ⊆ quotient of Cg by a lattice Λ Cg acting by translation. It is a complex manifold. ⊆ A complex torus Cg/Λ is a complex abelian variety if it admits an embedding in Pn for some n. 2. BACKGROUND 16

A complex torus Cg/Λ is an abelian variety if and only if there exists a positive deﬁnite Hermitian form1 H : Cg Cg C whose imaginary part is integral on Λ. × → (See [41, p. 28] for instance.) This fundamental result is related to the Appell-Humbert theorem which gives an explicit description of every line bundle on any complex torus. Appell-Humbert data and the Appell-Humbert theorem.

To formulate the Appell-Humbert theorem, we ﬁrst ﬁx some terminology. Let Λ Cg be a lattice and let H : Cg Cg C be a Hermitian form whose ⊆ × → imaginary part is integral on Λ. Let S1 := z C : z = 1 the set of complex { ∈ | | } numbers of length 1.

A semi-character for Λ with respect to H is a function χ : Λ S1 satisfying the → condition that

χ(λ + µ)= χ(λ)χ(µ)eπi Im H(λ,µ) for all λ, µ Λ. ∈ Appell-Humbert data, or simply AH-data, for Λ consists of a pair (H, χ) where

H : Cg Cg C × → is a Hermitian form whose imaginary part is integral on Λ and where χ is a semi- character for H with respect to Λ.

Example 2.5. Let Λ Cg be a lattice and let H : Cg Cg C be a Hermitian ⊆ × → form whose imaginary part is integral on Λ. We indicate how to construct a semi- character for H with respect to Λ. In fact, the collection of semi-characters for H

1 with respect to Λ is a HomZ(Λ, S )-torsor. (This is a consequence of the Appell- Humbert Theorem. See [6, Theorem 2.2.3, p. 32] for more details.)

1If V is a C-vector space then a Hermitian form is a map H : V V C which is C-linear in the × → ﬁrst component and has the property that H(v, w)= H(w, v), for all v, w V . ∈ 2. BACKGROUND 17

First, observe that H determines an alternating Z-bilinear form

Im H :Λ Λ Z. × →

Second, there exists positive integers (d ,...,d ), d d , and a decomposition 1 p i| i+1

Λ=Λ Λ Λ (2.3) 1 ⊕ 2 ⊕ 3

with the property that the matrix of Im H, with respect to the decomposition (2.3), takes block form 0 ∆ 0 0

 ∆ 0 0 0  −      0 0 0 0       0 0 0 0      where ∆ is the diagonal matrix having the positive integers (d1,...,dp) on the diagonal. (See [8, 5, Theorem 1, p. 79].) § Let χ :Λ S1 denote the map deﬁned by →

λ eπi Im H(λ1,λ2) 7→ where λ = λ + λ + λ and λ Λ for i =1, 2, 3. 1 2 3 i ∈ i If λ = λ + λ + λ ,µ = µ + µ + µ Λ and λ ,,µ Λ then 1 2 3 1 2 3 ∈ i i ∈ i

χ(λ + µ)= χ(λ)χ(µ)eiπ Im H(λ,µ)e−i2π Im H(λ2,µ1). (2.4)

Since Im H(λ2,µ1) is an integer, equation (2.4) implies that χ is a semi-character for H with respect to Λ.

Let (H, χ) be Appell-Humbert data for Λ. We now describe the manner in which (H, χ) determines a line bundle L(H, χ) on X := Cg/Λ. 2. BACKGROUND 18

The key point is that (H, χ) determines an action of Λ on Cg C. More precisely, × if (v, t) Cg C and λ Λ then ∈ × ∈

πH(v,λ)+ π H(λ,λ) λ (v, t):=(v + λ, tχ(λ)e 2 ). (2.5) ·

The quotient of Cg C with respect to (2.5) is the total space of a line bundle on X. × We denote this line bundle by L(H, χ); it is the line bundle determined by (H, χ).

The Appell-Humbert theorem asserts that–conversely–if L is a line bundle on X then L is isomorphic to L(H, χ) for some uniquely determined Appell-Humbert data (H, χ) for Λ. (See [41, p. 19] for instance.) The dual complex torus and the Poincare line bundle.

Let V := Cg, let Λ V be a lattice, and let X := V/Λ be a complex torus. We ⊆ ﬁrst describe its dual which is a complex torus X. We then describe the Poincare line bundle (which is a line bundle on X X having additional functorial properties) × b in terms of Appell-Humbert data. b

Let V := HomC(V, C) be the C-vector space of C-antilinear forms

b l : V C. →

The map , : V V R deﬁned by l, v := Im l(v) h− −i × → h i is a non-degenerate R-bilinearb form. In addition, the set

Λ := l V : l,λ Z for all λ Λ { ∈ h i ∈ ∈ } b b is a lattice in V so that the quotient X := V/Λ is a complex torus. We say that X is

the dual of X.b b b b 2. BACKGROUND 19

The map V Pic0(X) deﬁned by l e2πihl,−i induces an isomorphism → 7→

b 1 0 X = HomZ(Λ, S ) = Pic (X).

b To construct AH-data for the Poincare line bundle P on X X, we first note that × the map H :(V V ) (V V ) C defined by H((v , l ), (v , l )) = l (v )+ l (v ) × × × → 1 1 2 b2 2 1 1 2 determines a Hermitian form. Next, note that the map χ : Λ Λ S1 defined by b b × → χ(λ, l) := eπi Im l(λ) determines a semi-character for the lattice Λ Λ with respect to ×b H. The Poincare line bundle P is the line bundle L(H, χ) on X X. × b The Riemann-Roch formula and non-degenerate line bundles. b Let V := Cg, let Λ V be a lattice, and let X := V/Λ. We now describe the ⊆ Riemann-Roch formula for line bundles on X in terms of AH-data. Let (H, χ) be AH-data for Λ, and let L be the line bundle L(H, χ) on X. Let q denote the number of negative eigenvalues of H. Let Pf Im H denote the Pfaffian of (the alternating) form Im H :Λ Λ Z. × → In this notation, the Riemann-Roch formula becomes

χ(L(H, χ))=( 1)q Pf Im H. −

Now, let K(L) := x V : Im H(x, λ) Z for all λ Λ /Λ. We say that L is { ∈ ∈ ∈ } non-degenerate if K(L) is ﬁnite. We check that L is non-degenerate if and only if H

is non-degenerate. If L is non-degenerate, then let i(L) denote the number of negative eigenvalues of H. We have that Hb(X, L)=0 for b = i(L) and Hi(L)(X, L) =0. The number i(L) 6 6 is called the index of L. 2. BACKGROUND 20

2.4 Preliminary facts about line bundles on abelian

varieties

The main source for the material of this section is [41]. An abelian variety is a projective variety X, deﬁned over an algebraically closed ﬁeld k, with the structure of a group such that the maps (x, y) x + y and x x 7→ 7→ − are morphisms of algebraic varieties X X X and X X respectively. × → → Let X be an abelian variety. Central to the theory of line bundles on X is the subgroup Pic0(X) Pic(X) consisting of isomorphism classes of line bundles L ⊆ which satisfy the condition that T ∗L = L for all x X. x ∼ ∈ There exists an abelian variety X, the dual abelian variety, and a line bundle P

on X X with the properties that × b (a) if x X, then P := P is an element of Pic0(X); b∈ x |X×{x} ∼ (b) the map x P bdeﬁnes anb isomorphism of groups X = Pic0(X); b b 7→ x −→ (c) P = andb P = . |X×{0}∼b OX |{0}×X ∼ OX b The line bundle P is called the (normalized)b b Poincare line bundle.

For every line bundle L on X, there is a unique homomorphism

φ : X X L → b characterized by the property that P = T ∗(L) L−1. φL(x) ∼ x ⊗

The k-valued points of kernel of φL is the group

K(L) := x X : T ∗L = L . { ∈ x ∼ }

If K(L) is ﬁnite, then we say that L is non-degenerate. 2. BACKGROUND 21

The Riemann-Roch theorem for abelian varieties asserts that

1 g χ(L)= c1(L) =+ deg φL. g! X Z p Mumford proved that a line bundle L on X is non-degenerate if and only if χ(L) =0. In addition, if L is a non-degenerate line bundle on X, then 6 (a) there exists an integer i(L), the index of L, with the property that Hb(X, L)=0

for b = i(L); 6 (b) if A is an ample line bundle on X and if χ(L) = 0 then the roots of the poly- 6 nomial P (N) := χ(AN L) are real and i(L) equals the number of positive ⊗ roots counted with multiplicity.

See [41, 16]. §

2.5 Preliminary facts about vector bundles on abelian

varieties

Atiyah proved that, for a fixed rank r and fixed degree d, the set of equivalence classes of indecomposable2 vector bundles of rank r and degree d, on an elliptic curve X, can be identified with X [5, Theorem 7, p. 434]. These results were extended by T. Oda [43], [42], and T. Matsushima [31]. Mukai also considered the case of higher rank vector bundles on abelian varieties X of higher dimension. He generalized many aspects of the theory of line bundles on X to vector bundles on X [33]. He gave several definitions which cap- ture various properties of line bundles.

2Let X be an abelian variety. A vector bundle E on X is said to be decomposable if it can be written as a direct sum of proper sub-bundles. If E is not decomposable then we say that E is indecomposable. 2. BACKGROUND 22

For instance, let E be a vector bundle on X

E is homogeneous if T ∗(E) = E for all x X; • x ∼ ∈ E is simple if End (E) = k; • OX ∼ E is semi-homogeneous if for every x X there exists a line bundle L on X • ∈ such that T ∗(E) = E L. x ∼ ⊗ In addition, Mukai constructed simple semi-homogeneous vector bundles with

a speciﬁed slope [33, 6]. More precisely, if E is a vector bundle on X, then let r(E) § denote its rank and let slope(E) denote the class of det(E) in NS(X) Q. Mukai r(E) ⊗ established that, for every d NS(X) Q, there exists a simple semi-homogeneous ∈ ⊗ vector bundle E on X with slope(E)= d, [33, Corollary 6.23]. (Compare also with

the approach of Matsushima [31].) Mukai also proved that if E is a simple semi-homogeneous vector bundle on X then (a) dim K(E) = dim K(det E); (b) if χ(E) =0 then #K(E) < and #K(E)= χ(E)2. 6 ∞ See [33, Corollary 7.9, p. 271].

2.6 Other notation and conventions

Throughout (unless explicitly stated otherwise) all abelian varieties are defined over a fixed algebraically closed field k of arbitrary characteristic.

If E is a vector bundle on a g-dimensional abelian variety X then we let cℓ(E)

ℓ denote its ℓth Chern class in the group A (X) = Ag−ℓ(X) of codimension ℓ-cycles 2. BACKGROUND 23

modulo rational equivalence. We let chg(E) denote the component of ch(E) con-

g tained in A (X)Q. In this notation, the Hirzebruch-Riemann-Roch theorem reads

χ(E)= chg(E) ZX g where if x = n [p] A (X)Q then x denotes the number n . We refer to p p ∈ X p p [13] for moreP details regarding intersectionR theory. P

Let E := Ep,q,dp,q be a spectral sequence arising from a ﬁltered complex of { r r } n k-vector spaces. Let M Z be a collection of k-vector spaces with the property { }n∈ that each M n admits a descending ﬁltration

F p−1M n F pM n F p+1M n , ···⊇ ⊇ ⊇ ⊇···

with F qM n = M n, for all sufficiently small q, and F qM n = 0, for all sufficiently large q. We use the notation Ep,q M p+q r ⇒ to mean that, for all sufficiently large integers s, and all integers p and q, we have specified (split) short exact sequences

0 F p+1M p+q F pM p+q Ep,q 0. → → → s → 3

Curves: Segre embeddings and canonical curves

We prove that there is no g for which the canonical embedding of a general curve of genus g lies on the Segre embedding of any product of three or more projective spaces. The results of this chapter depend on Theorem 2.1 (a), Theorem 2.3, and Theorem 2.4 (a).

3.1 Statement of results and outline of their proof

To prove our main result, we ﬁrst give the following criterion for a general curve to lie on some Segre embedding Pr1 Prn Pg−1. ×···× →

Proposition 3.1. The canonical image of a general curve C of genus g lies on some Segre

r1 rn g−1 embedding P P P if and only if C admits line bundles L1,...,Ln such n ×···× → n 0 that Li = KC , h (C, Li)= ri +1, and (ri +1)= g. i=1 i=1 N Q 24 3. CURVES: SEGRE EMBEDDINGS AND CANONICAL CURVES 25

We then prove the following stronger result of independent interest.

Theorem 3.2. Let C be a general curve of genus g and let n > 3. If C admits line bundles n 0 L1,...,Ln such that deg Li = 2g 2 and h (C, Li) = ri +1 > 2 for i = 1,...,n, i=1 − then n =3 and P

3 3 r + r + r +2 (r + 1) < (r + 1) 1 2 3 6 g. i i r + r + r +2 r r r i=1 i=1 1 2 3 1 2 3 Y Y − Moreover, up to permutation of the ri, we have (a) r = r =1 and 1 6 r 6 g 2 or 1 2 3 4 −

(b) r1 =1, r2 =2 and r3 =2, 3, or 4.

A more conceptual interpretation of the theorem is the following corollary.

Corollary 3.3. Let C be a general curve of genus g.

(a) If g is composite, then the canonical image of C lies on some Segre embedding

Pr1 Pr2 Pg−1. × →

(b) If n > 3, then the canonical image of C does not lie on any Segre embedding

Pr1 Prn Pg−1. ×···× →

Statement (a) in Corollary 3.3 is well-known to experts and follows from Brill- Noether theory. We include a proof for completeness. Throughout, we assume that g is an integer greater than or equal to 3.

3.2 The n-fold Petri-map and proof of Proposition 3.1

Proposition 3.1 relies on the following lemma which implies that, for a general curve, the n-fold Petri-map is injective. 3. CURVES: SEGRE EMBEDDINGS AND CANONICAL CURVES 26

Lemma 3.4. Let C be a general curve of genus g. For all line bundles L1,...,Ln on C n n 0 0 such that Li ∼= KC , the cup-product H (C, Li) H (C,KC) is injective. i=1 i=1 → N N Proof. The case n =1 is trivial. The case n =2 is the Gieseker-Petri theorem (Theo- rem 2.3). Let n > 3. Without loss of generality we may assume that H0(C, L ) =0, i 6 i =1,...n. The cup-product factors

0 0 0 H (C, L1) H (C, Ln) / H (C,KC) . ⊗···⊗ gggg3 ggggg ggggg ggggg ggg H0(C, L L ) H0(C, L ) H0(C, L ) 1 ⊗ 2 ⊗ 3 ⊗···⊗ n By induction, the diagonal arrow is injective. It thus sufﬁces to show that the

downward arrow is injective. For this, we reduce to showing that the cup-product

Φ : H0(C, L ) H0(C, L ) H0(C, L L ) 1 ⊗ 2 → 1 ⊗ 2

is injective.

By assumption, there exist non-zero sections σ H0(C, L ), i = 3,...n. These i ∈ i produce (via cup-product) a non-zero section

σ = σ σ H0(C, L L ). 3 ··· n ∈ 3 ⊗···⊗ n

Since σ =0 multiplication by σ yields the following two injections 6

H0(C, L ) H0(C, L L ) 2 → 2 ⊗···⊗ n

and H0(C, L L ) H0(C,K ). 1 ⊗ 2 → C 3. CURVES: SEGRE EMBEDDINGS AND CANONICAL CURVES 27

Using the above we produce (by cup-product) the commutative diagram

Φ H0(C, L ) H0(C, L ) / H0(C, L L ) . 1 ⊗ 2 1 ⊗ 2 id ⊗·σ ·σ

µ(L1) H0(C, L ) H0(C, L L ) / H0(C,K ) 1 ⊗ 2 ⊗···⊗ n C

The vertical arrows of the above diagram are injective, as just noted, whereas bottom arrow of the diagram is injective by the Gieseker-Petri theorem. Hence, the top arrow Φ is injective.

We now use Lemma 3.4 to prove Proposition 3.1.

Proof of Proposition 3.1. Let η : C Pg−1 be the canonical map. If C is contained in → the image of some Segre embedding φ : Pr1 Prn Pg−1 then there exists a ×···× → closed immersion ψ : C Pr1 Prn making the diagram → ×···×

η C / g−1 o7 P ooo ψ ooo ooo φ oo Pr1 Prn ×···× commute.

For every 1 6 i 6 n, let π denote the projection of Pr1 Prn onto the i-th i ×···× ∗ ∗ ∗ factor and set L := (π ψ) Pri (1) = ψ (π Pri (1)). We then obtain i i ◦ O ∼ i O

∗ ∗ K = η Pg−1 (1) = (φ ψ) Pg−1 (1) = L L . C ∼ O ∼ ◦ O ∼ 1 ⊗···⊗ n

Since the canonical image of C is non-degenerate we conclude that

0 h (C, Li) > ri +1 for i =1,...,n. 3. CURVES: SEGRE EMBEDDINGS AND CANONICAL CURVES 28

Finally, since η is induced by the complete canonical series, we conclude that n 0 0 the cup-product H (C, Li) H (C,KC) is surjective. By Lemma 3.4 the cup- i=1 → N n product is injective and, by assumption, (ri +1) = g. It follows that i=1 Q 0 h (C, Li)= ri +1 for i =1,...,n.

Conversely, given such L ,...,L we get regular maps C Pri for i =1,...,n. 1 n → We thus can make a regular map η : C Pg−1 (induced by a (sub)-canonical series) → Segre by composition C / Pr1 Prn / Pg−1 . By Lemma 3.4, the cup-product n ×···× n 0 0 H (C, Li) H (C,KC) is injective. Since (ri +1) = g, it is also surjective. We i=1 → i=1 thusN conclude that the resulting map η is givenQ by the complete canonical series.

3.3 Proof of Theorem 3.2 and its corollary

We ﬁrst prove Corollary 3.3 (a). We then prove Theorem 3.2 from which we deduce Corollary 3.3 (b).

The case n =2 and g is composite

When n =2 and g is composite, it is easy to prove that, in its canonical embedding, a general curve of genus g lies on the image of some (non-trivial) Segre embedding

Pr1 Pr2 Pg−1. × →

Proof of Corollary 3.3 (a). Since g is composite, we can write g =(r1 +1)(r2 +1) with r > 1. Set d = r1g + r = r r +2r . Then ρ(g,r ,d )=0 so, by Theorem 2.1 (a), i 1 r1+1 1 1 2 1 1 1 3. CURVES: SEGRE EMBEDDINGS AND CANONICAL CURVES 29

0 C admits a line bundle L1 with h (C, L1) > r1 +1. On the other hand,

ρ(g,r1 +1,d1) < 0

r1+1 0 so Theorem 2.4 (a) implies that Wd1 = ∅. Hence, h (C, L1)= r1 +1. Set

L := L∨ K . 2 1 ⊗ C

Then, by the Riemann-Roch theorem, we obtain h0(C, L )= g d +r . Simplifying, 2 − 1 1 and using our expressions for g and d1 above, we obtain

h0(C, L )=(r + 1)(r + 1) r r 2r + r = r +1. 2 1 2 − 1 2 − 1 1 2

The assertion now follows from Proposition 3.1.

The case n > 3

When n > 3 and ri > 1, for i = 1,...,n, the situation differs from the case n = 2. Indeed, we prove that there is no g for which the canonical embedding of a general curve lies on the Segre embedding of any product of three or more projective spaces. We deduce this result from Theorem 3.2 whose proof occupies the rest of

this section. The theorem follows from the following more general observation.

Proposition 3.5. Let C be a general curve of genus g. Let d be a non-negative integer and d 0 > let q := g . If C admits line bundles L1,...,Ln such that ri +1= h (C, Li) 2 and n n j k 1+ ri „i=1 « deg Li = d, then n< 2q +2 and g > P n . i=1 −n+(q+1)+ 1 „ ri+1 « N iP=1 Proof. Since C is general, if such Li exist then, by Theorem 2.4 (a),

ρ(g,r ,d )= g (r + 1)(g + r d ) > 0 for i =1,...,n. i i − i i − i 3. CURVES: SEGRE EMBEDDINGS AND CANONICAL CURVES 30

g Solving for di, we conclude di > g + ri for i = 1,...,n. Let r denote the − ri+1 remainder obtained by dividing d by g. Then 0 6 r

n n n g d =(q + 1)g + r g = d > ng + r . − i i − r +1 i=1 i=1 ! i=1 i ! X X X Rearranging we obtain

n 1 n n (q + 1) g 6 r g r < 0. (3.1) − − r +1 − − i i=1 i !! i=1 ! X X

Since g > 0 we conclude

n 1 n (q + 1) < 0. (3.2) − − r +1 i=1 i ! X Now by assumption ri > 1, for i =1,...,n. Thus

n 1 n 6 . (3.3) r +1 2 i=1 i X Using equations (3.3) and (3.2) we deduce that n < 2q +2. Finally, dividing equation (3.1) by equation (3.2), we obtain

n n r + g + r 1+ r − i i g > i=1 > i=1 . (3.4) Pn P n n +(q +1)+ 1 n +(q +1)+ 1 − ri+1 − ri+1 i=1 i=1 P P

We now use Proposition 3.5 to prove Theorem 3.2.

Proof of Theorem 3.2. Set d = deg L , i =1,...,n and set d =2g 2. Then i i − d q := =1 g j k 3. CURVES: SEGRE EMBEDDINGS AND CANONICAL CURVES 31

and the remainder r equals g 2. Since r > 1, for i =1,...,n, applying Proposition − i 3.5 we conclude that n< 4. Since n > 3 we conclude that n =3. Substituting n =3 and r = g 2 into equation (3.4) we obtain − 3 2+ ri g > i=1 . P3 1+ 1 ri+1 − i=1 P Rearranging, we obtain

3 r + r + r +2 3 g > (r + 1) 1 2 3 > (r + 1). i r + r + r +2 r r r i i=1 ! 1 2 3 1 2 3 i=1 Y − Y

3 Since 1 < 1 we conclude, up to permutation of the r , that r = r =1, r > 1 ri+1 i 1 2 3 i=1 or r1 =1,rP2 =2 and 2 6 r3 6 4. Finally if r1 = r2 =1 and r3 > 1 then the condition 3 6 6 g (ri + 1)

Proof of Corollary 3.3 (b). Let C be a general curve of genus g. Suppose that C lies

on the image of some Segre embedding Pr1 Prn Pg−1. By Proposition ×···×g → 0 3.1, C admits line bundles L1,...,Ln such that Li = KC , h (C, Li) = ri +1 and i=1 n 3 N (ri +1) = g. By Theorem 3.2, n =3 and g > (ri +1). This is a contradiction. i=1 i=1 Q Q 4

Abelian varieties: index conditions and cup-products

We study a cohomological condition which we place on a pair of vector bundles

on an abelian variety. We determine the asymptotic nature of this condition and use this result to ascertain the behaviour of families of cup-product problems. One consequence of our main result is that vector bundles, arising from families of cup- product maps, are asymptotically globally generated.

4.1 The pair index condition

Let X be an abelian variety defined over an algebraically closed field. We define a cohomological condition which we place on a pair of vector bundles on X. To state this criterion, we first make some auxiliary definitions. Definitions. Let E be a vector bundle E on X.

We say that E is non-degenerate if χ(E) =0. • 6 32 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 33

If E admits exactly one nonzero cohomology group, then we say that E sat- • isﬁes the index condition (or IC).

If E satisfies the IC, then let i(E) denote the unique integer b for which • Hb(X, E) is nonzero and say that i(E) is the index of E. These definitions are motivated by the theory of line bundles on X. For instance, D. Mumford proved that a line bundle on X satisfies the IC if and only if it is non-degenerate [41, 16, p. 140–143]. § If x X then T : X X denotes translation by x in the group law. ∈ x → Definition. A pair (E, F ) of vector bundles on X satisfies the pair index condition (or PIC) if, for all x X, the vector bundles T ∗E, F , and T ∗(E) F satisfy the IC ∈ x x ⊗ and i(T ∗(E) F )=i(T ∗E)+i(F ). x ⊗ x Every pair (E, F ) of vector bundles on X which satisfies the PIC determines cup-product maps

∗ ∪ ∗ (T ∗E, F ) : Hi(Tx E)(X, T ∗E) Hi(F )(X, F ) Hi(Tx (E)⊗F )(X, T ∗(E) F ), (4.1) ∪ x x ⊗ −→ x ⊗ for all x X, with nonzero source and target space. ∈ Remark. Using upper-semicontinuity we deduce a pair (E, F ) of vector bundles on X satisﬁes the PIC if and only if the vector • bundles T ∗E, F , and T ∗(E) F satisfy the IC, for all x X, and x x ⊗ ∈ i(E)+i(F )=i(E F ); ⊗ a pair (L, M) of line bundles on X satisﬁes the PIC if and only if L, M, and • L M are non-degenerate and i(L)+i(M)=i(L M). ⊗ ⊗ See [41, part (a) of the corollary on p. 47] for instance. We study the PIC for a certain class of complex abelian varieties with real multiplication in 5.2. § 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 34

4.2 Statement of results and outline of their proof

Our ﬁrst result concerns the asymptotic nature of the PIC.

Theorem 4.1. Let X be an abelian variety and let (L, M) be a pair of line bundles on X satisfying the PIC. If E and F are vector bundles on X, then there exists a positive integer n such that, for all n > n , the pair (Ln E, M n F ) satisﬁes the PIC and 0 0 ⊗ ⊗

i(T ∗(Ln E)) = i(L), i(M n F )=i(M), and i(T ∗(Ln E) M n F )=i(L M), x ⊗ ⊗ x ⊗ ⊗ ⊗ ⊗ for all x X. ∈ There are two important points to our proof of Theorem 4.1. The ﬁrst is our interpretation of Mumford’s index theorem which we relate to the real Neron-Severi space of X. (See 4.3.) The second is the following theorem which also implies that § if L is a non-degenerate line bundle on X and q := i(L) then L is q-ample. (See

Corollary 4.13.)

Theorem 4.2. Let Y be an abelian variety. Let L and F denote, respectively, a non- degenerate line bundle and a coherent sheaf on Y . There exists a positive integer n0 such that, for all isogenies f : X Y , we have Hj(X, f ∗(F Ln) α)=0 for all j > i(L), → ⊗ ⊗ for all n > n , and for all α Pic0(X). 0 ∈ Let p denote the projection of X X onto the ﬁrst and second factors respec- i × tively (for i =1, 2) and let m := p1 + p2. We prove, in Proposition 4.8, that the image of the cup-product (4.1), for x X, ∈ coincides with the ﬁberwise evaluation map

H0(X, Ri(E⊗F )(m∗E p∗F )) κ(x) Ri(E⊗F )(m∗E p∗F ) p1∗ ⊗ 2 ⊗ → p1∗ ⊗ 2 |x 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 35

of the vector bundle Ri(E⊗F )(m∗E p∗F ) on X. p1∗ ⊗ 2 Using this relationship, Theorem 4.1, Theorem 4.2, and results of Pareschi- Popa, [44], [45] and [46], we give two proofs of the main result of this chapter. (See 4.5.4 and 4.5.5.) § § Theorem 4.3. Let X be an abelian variety, let E and F be vector bundles on X, and let

(L, M) be a pair of line bundles on X satisfying the PIC. There exists a positive integer n0 such that the cup-product

∪ Hi(L)(X, T ∗(Ln E)) Hi(M)(X, M n F ) Hi(L⊗M)(X, T ∗(Ln E) M n F ) (4.2) x ⊗ ⊗ ⊗ −→ x ⊗ ⊗ ⊗ is nonzero and surjective for all n > n and all x X. Equivalently, the vector bundle 0 ∈

Ri(L⊗M)(m∗(Ln E) p∗(M n F )) (4.3) p1∗ ⊗ ⊗ 2 ⊗ is globally generated if n > n0.

Our ﬁrst approach to proving Theorem 4.3 is logically independent of the main results of [44], [45], and [46].

In more detail, using Theorems 4.2 and 4.1, we prove that the cup-product maps (4.2) are nonzero and surjective, for all x X, and all sufﬁciently large n. (See ∈ Proposition 4.15.) Applying Proposition 4.8, which generalizes [44, Proposition 2.1], we deduce that the vector bundles (4.3) are globally generated whenever n is sufﬁciently large.

Our second approach to proving Theorem 4.3 is to prove that the vector bundles (4.3) are globally generated whenever n is sufﬁciently large. We then deduce, using Proposition 4.8, that the cup-product maps (4.2) are nonzero and surjective, for all points of X, and all sufﬁciently large n. 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 36

In our second approach, to prove that the vector bundles (4.3) are globally generated, we apply Pareschi-Popa’s theory of M-regularity. Speciﬁcally, we apply

[45, Theorem 2.4, p. 289] to obtain a sufﬁcient condition for such vector bundles to be globally generated. (See Proposition 4.16.) In fact, we do not need the full strength of their theory [44, Theorem 2.1, p. 654] sufﬁces. Our second approach then proves Theorem 4.3 by combining Proposition 4.16 with Theorem 4.1.

4.3 Mumford’sindex theoremand the real Neron-Severi

space

Let X be an abelian variety of dimension g and let N1(X) denote the group of line bundles on X modulo numerical equivalence. We relate Mumford’s index theorem

1 1 N (X)R := N (X) Z R ⊗ the real Neron-Severi space of X. (See also work of A. K¨uronya [25].) Mumford’s proof of this theorem, see [41, p. 145–152], was later extended by

G. Kempf and, independently, by C.P. Ramanujam [39, Appendix]. More recently, B. Moonen and G. van der Geer have given a very clear exposition of Mumford’s proof [32, p. 134-139]. 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 37

4.3.1 Non-degenerate R-divisors

If L and M are numerically equivalent line bundles, then they have equal Euler characteristic [22, Theorem 1, p. 311]. We thus have a well deﬁned function

χ : N1(X) Z (4.4) →

deﬁned by sending the numerical class of a line bundle to its Euler characteristic. Using the Riemann-Roch theorem

1 1 χ(L)= c (L)g = (Lg), g! 1 g! ZX where L is a line bundle on X and (Lg) denotes its g-fold self-intersection number,

we deduce that the function (4.4) extends to a function

1 χ : N (X)R R (4.5) → by extending scalars.

Note that χ is a homogeneous polynomial function of degree g. Indeed, let x1,...,xm be numerical classes of line bundles L1,...,Lm which form a basis for N1(X). If m x = a x , with a R, i i i ∈ i=1 X then χ(x) is the value of the polynomial

c (L )l1 . . . c (L )lm X 1 1 1 m l1 lm P (X1,...,Xm) := X1 ...Xm l1! ...lm! l1+···+lm=g,li>0 R X

evaluated at a1,...,am.

1 We say that x N (X)R is non-degenerate if χ(x) =0. Since χ is a homogeneous ∈ 6 1 polynomial function its non-vanishing determines an open subset of N (X)R. 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 38

1 4.3.2 The index of non-degenerate elements of N (X)Q

If x N1(X) is non-degenerate, then we deﬁne i(x) := i(L), where L is any line ∈ bundle with numerical class equal to x. Mumford’s index theorem implies that numerically equivalent non-degenerate line bundles have the same index so this is well-deﬁned.

1 If x is a non-degenerate element of N (X)Q, then so is ax for all nonzero integers a. In addition ax N1(X), for some positive integer a, and we deﬁne ∈

i(x) := i(ax).

Since i(L) = i(Ln), for all non-degenerate line bundles L and all positive integers n, see for instance [41, Corollary p. 148], this is well-deﬁned.

1 If x N (X)Q is non-degenerate, then we refer to i(x) as the index of x. ∈

1 4.3.3 The index function and the non-degenerate locus of N (X)R

1 Proposition 4.4. Let U be the subset of N (X)R deﬁned by the non-vanishing of χ. If x

1 and y are elements of N (X)Q and lie in the same connected component of U, then they have the same index.

Proof. Let S be a connected component of U and suppose that x and y are elements

1 m of N (X)Q lying in S. Since S is an open subset of R , it is path connected so there exists a path from x to y. This path can be approximated by straight line segments, each of which is contained in S, of the form tz + (1 t)w, where t [0, 1], and z − ∈ 1 and w are elements of N (X)Q lying in S. To prove Proposition 4.4, it sufﬁces to prove that if x and y are elements of

1 N (X)Q, lying in S, and connected by a straight line, tx+(1 t)y, t [0, 1], contained − ∈ 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 39

in S then i(x)=i(y). By scaling the straight line, we reduce further to showing that if x and y are

elements of N1(X) and connected by a straight line, tx +(1 t)y, t [0, 1], lying in − ∈ S then x and y have the same index. This is precisely Step B in Mumford’s index theorem see [41, p. 147]. (Note the typo in the third line of Step B in the reprinted edition. It should read F (t, 1 t) =0 rather than F (t, 1 t)=0. Compare with the − 6 − original.)

1 Corollary 4.5. The index function i : U N (X)Q 0,...,g extends to a continuous ∩ →{ } function i : U 0,...,g . →{ }

Proof. If x is an element of U then it is contained in a connected component S of

1 1 U. Since N (X)Q is dense in N (X)R and since S is open we may deﬁne i(x) := i(y)

1 where y is some element of N (X)Q S. Proposition 4.4 implies that this is well- ∩ deﬁned. This function is constant on the connected components of U and hence is

continuous.

The following consequence of Proposition 4.4 plays a central role in our proof of Theorem 4.2.

Corollary 4.6. Let x be a non-degenerate integral class and let y be an integral class. There exists a positive integer a such that χ(ax + y) =0 and i(ax + y)=i(x) for all a > a . 0 6 0

Proof. Let S be a connected component of U containing x. Since S is open there exists an open ball B around x and contained in S. Then x + 1 y B for all ǫ a ∈ ǫ 1 sufﬁciently large integers a. Since ax + y = a(x + a y), we conclude that

1 χ(ax + y) =0 and i(ax + y)=i(x + y)=i(x), 6 a 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 40

for all sufﬁciently large integers a.

4.4 Families of cup-product problems

Let X be an abelian variety and let (E, F ) be a pair of vector bundles on X satisfying the PIC. In this section, we relate the cup-products

∗ ∗ (T ∗E, F ) : Hi(Tx E)(X, T ∗E) Hi(F )(X, F ) Hi(Tx (E)⊗F )(X, T ∗(E) F ), ∪ x x ⊗ → x ⊗ for x X, to the fiberwise evaluation map ∈ H0(X, Ri(E⊗F )(m∗E p∗F )) κ(x) Ri(E⊗F )(m∗E p∗F ) , p1∗ ⊗ 2 ⊗ → p1∗ ⊗ 2 |x i(E⊗F ) ∗ ∗ of the sheaf Rp ∗ (m E p F ) on X. 1 ⊗ 2 i(E⊗F ) ∗ ∗ Our first proposition clarifies the nature of the sheaf Rp ∗ (m E p F ). 1 ⊗ 2 Proposition 4.7. Let (E, F ) be a pair of vector bundles on X satisfying the PIC. The sheaf

Ri(E⊗F )(m∗E p∗F ) p1∗ ⊗ 2 is a rank χ(E F ) vector bundle on X. In addition, we have | ⊗ | H0(X, Ri(E⊗F )(m∗E p∗F ))=Hi(E⊗F )(X X, m∗E p∗F ) p1∗ ⊗ 2 × ⊗ 2 and Hj(X, Ri(E⊗F )(m∗E p∗F ))=0, for j > 0. p1∗ ⊗ 2 Proof. Let := m∗E p∗F and let := Ri(E⊗F )( ). Note that is ﬂat over X via N ⊗ 2 E p1∗ N N p . Since the vector bundles = T ∗(E) F , for x X, satisfy the IC, and 1 N |{x}×X x ⊗ ∈ have index i(E F ), we have that, for j Z, ⊗ ∈ χ(T ∗(E) F ) when j = i(E F ) j x dimκ(x) H ( x X, {x}×X )= | ⊗ | ⊗ (4.6) { } × N |  0 when j = i(E F ). 6 ⊗   4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 41

Since the Euler characteristic of a ﬂat family of sheaves over a connected base is constant, we see that, for j ﬁxed, the function x dim Hj( x X, ) is 7→ κ(x) { } × N |{x}×X constant. This is condition (i) of [41, Cor. 3, p. 40] which is equivalent to the condition that Rj ( ) is a vector bundle and that the natural map p1∗ N

Rj ( ) Hj( x X, ) p1∗ N |x→ { } × N |{x}×X is an isomorphism. Using (4.6) we conclude that is a vector bundle of the asserted E rank and, moreover, if j = i(E F ) then Rj ( )=0. 6 ⊗ p1∗ N To compute the cohomology groups of , we use the Leray spectral sequence E

Ep,q = Hp(X, Rq ( )) Hp+q(X X, ). 2 p1∗ N ⇒ × N

Since Rj ( )=0, when j = i(E F ), we have Hℓ−i(E⊗F )(X, ) = Hℓ(X X, ), for p1∗ N 6 ⊗ E × N all ℓ. Notably H0(X, ) = Hi(E⊗F )(X X, ) while the higher cohomology groups E × N of are zero because the cohomology groups of are zero when j = i(E F ). E N 6 ⊗

Our next proposition generalizes [44, Proposition 2.1] (see also [46, Proposition 5.2]), where it plays a central role in G. Pareschi’s proof of Lazarsfeld’s conjecture.

(See [44, p. 660–663] and also [46, 6].) § Proposition 4.8. Let (E, F ) be a pair of vector bundles on X satisfying the PIC. The image of the cup-product

∗ ∗ (T ∗E, F ) : Hi(Tx E)(X, T ∗E) Hi(F )(X, F ) Hi(Tx (E)⊗F )(X, T ∗(E) F ), ∪ x x ⊗ → x ⊗ for all x X, coincides with the image of the ﬁberwise evaluation map ∈

H0(X, Ri(E⊗F )(m∗E p∗F )) κ(x) Ri(E⊗F )(m∗E p∗F ) . (4.7) p1∗ ⊗ 2 ⊗ → p1∗ ⊗ 2 |x 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 42

∗ ∗ i(E⊗F ) Proof. Let := m E p F and let := Rp ∗ ( ). If x X then, using Proposi- N ⊗ 2 E 1 N ∈ tion 4.7, the isomorphisms

(X X, T 0,p )=(X X,p ,p )=(X X,m,p ), × x × 2 × 1 2 × 2 and repeated application of the K¨unneth formula, we obtain the commutative diagram

eval H0(X, ) κ(x) / E ⊗ E|x

i(E⊗F ) res i(T ∗(E)⊗F ) H (X X, ) / H x ( x X, {x}×X ) × N { } × N |

∗ ∪ ∗ Hi(Tx E)(X, T ∗E) Hi(F )(X, F ) // Hi(Tx (E)⊗F )(X, T ∗(E) F ) x ⊗ x ⊗ from which the assertion follows.

Corollary 4.9. Let (E, F ) be a pair of vector bundles on X satisfying the PIC.

(a) There exists a point x of X for which the cup-product (T ∗E, F ) is nonzero. ∪ x (b) The cup-products (T ∗E, F ) are nonzero and surjective, for all x X, if and only ∪ x ∈ if the vector bundle Ri(E⊗F )(m∗E p∗F ) p1∗ ⊗ 2 is globally generated.

Proof. To prove (a), using the K¨unneth formula and Proposition 4.7, we deduce that Ri(E⊗F )(m∗E p∗F ) admits a nonzero global section. The zero locus of this p1∗ ⊗ 2 section is a proper closed subset. Consequently, if x is not in this closed subset then using, Proposition 4.8, we deduce that (T ∗E, F ) is nonzero. ∪ x 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 43

Part (b) is also a consequence of Proposition 4.8: if x X then the image of ∈ (T ∗E, F ) coincides with the ﬁberwise evaluation map (4.7). Hence, if (T ∗E, F ) ∪ x ∪ x i(E⊗F ) ∗ ∗ is surjective, for all x X, then so is (4.7) so that Rp ∗ (m E p F ) is globally ∈ 1 ⊗ 2 generated. Conversely, suppose that the vector bundle Ri(E⊗F )(m∗E p∗F ) is globally gen- p1∗ ⊗ 2 erated. Then, since it has positive rank, the cup-products (T ∗E, F ), for all x X, ∪ x ∈ i(E⊗F ) ∗ ∗ are nonzero. In addition, since Rp ∗ (m E p F ) is globally generated, the ﬁber- 1 ⊗ 2 wise evaluation map (4.7) is surjective. Hence (T ∗E, F ) is surjective. ∪ x To prove (c) if Ri(E⊗F )(m∗E p∗F ) is trivial then Hg(X, Ri(E⊗F )(m∗E p∗F )) =0 p1∗ ⊗ 2 p1∗ ⊗ 2 6 which contradicts Proposition 4.7 when g is positive.

4.5 Proof of theorems

In 4.5.1, we make two remarks which we use to prove Theorems 4.1, 4.2, and § 4.3. In 4.5.2, we prove Theorem 4.2 and state two consequences, one of which we § use to establish Theorem 4.1. We prove Theorem 4.1 in 4.5.3. Our ﬁrst proof of § Theorem 4.3 is contained in 4.5.4 while our second and its relation to the work of § Pareschi-Popa (see [45] and [46]) is contained in 4.5.5. §

4.5.1 Two preliminary remarks

Let X be a projective variety of dimension g and let A be a globally generated ample line bundle on X. A coherent sheaf F on X is said to be m-regular with respect to A if Hi(X, F A⊗m−i)=0 for all i> 0. (See for example [35, Lecture 14], ⊗

[22, p. 307], or [28, Deﬁnition 1.8.4].) We deﬁne regA(F ) to be the least integer for 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 44

which F is m-regular with respect to A. One feature of m-regularity is that it controls the shape of a resolution of a coherent sheaf.

Proposition 4.10 (See also [1, Corollary 3.2, p. 240]). Let A be a globally generated ample line bundle on a g-dimensional projective variety X and set p := max 1, reg ( ) . { A OX } If a coherent sheaf F on X is m-regular with respect to A then there exists an exact complex of sheaves 0 F F F F 0 → g+1 → g →···→ 0 → →

−pℓ−m where, for 0 6 ℓ 6 g, Fℓ is a ﬁnite direct sum of copies of A .

Proof. Using, for example [22, Proposition 1, p. 307] or [28, Theorem 1.8.5, p. 100], we observe that, if F is m-regular with respect to A, then the kernel of the evaluation map H0(X, F A⊗m) A⊗−m F ⊗ ⊗ → is p+m-regular. Using this fact, the proposition follows easily using induction.

The following lemma relates the vanishing of certain terms on the E1 page of an appropriate spectral sequence to the vanishing of a particular cohomology group of a sheaf on a projective variety.

Lemma 4.11 (See also [1, Lemma 2.1, p. 235]). Let E be a vector bundle, let F be a coherent sheaf, and let : 0 F F F F 0 be an exact complex F → g+1 → g →···→ 0 → → of sheaves on a g-dimensional projective variety X. If Hℓ+q(X, E F )=0, whenever ⊗ ℓ 0 6 ℓ 6 g q, then Hq(X, E F )=0. − ⊗ 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 45

Proof. Since E is a vector bundle the complex E is also exact. Associated to ⊗F the complex E is a spectral sequence which has ⊗F

i H (X, E Fℓ) for 0 6 ℓ 6 g +1 and i Z −ℓ,i ⊗ ∈ i−ℓ E1 =  H (X, F E).  ⇒ ⊗ 0 for ℓ>g +1 or ℓ< 0 and i Z ∈  Since E−ℓ,i is zero, whenever i ℓ = q, we conclude that Hq(X, E F ) is zero as 1 − ⊗ well.

4.5.2 Proof and discussion of Theorem 4.2

We now use 4.3 and 4.5.1 to prove Theorem 4.2. § §

Proof of Theorem 4.2. Fix a globally generated ample line bundle A on Y . Further-

more, let m := regA(F ), let q := i(L), and let p := g +1. By Corollary 4.6, there exists a positive integer n0 with the property that

χ(A−pℓ−m Ln) =0 and i(A−pℓ−m Ln)= q, (4.8) ⊗ 6 ⊗ for all n > n0 and all 0 6 ℓ 6 g. Fix such an n , let n > n , let f : X Y be an isogeny, and let α Pic0(X). 0 0 → ∈ Using (4.8) and Mumford’s index theorem, see also [32, Cor. 9.26, p. 140], we deduce that

χ(f ∗(A−pℓ−m Ln) α) =0 and i(f ∗(A−pℓ−m Ln) α)= q, (4.9) ⊗ ⊗ 6 ⊗ ⊗ for all n > n0 and all 0 6 ℓ 6 g. (Note that f need not be a separable isogeny.) Using (4.9) we conclude that

Hℓ+j(X, f ∗(A−pℓ−m Ln) α)=0, (4.10) ⊗ ⊗ 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 46

whenever ℓ + j = q. In particular (4.10) holds for all j > q and all 0 6 ℓ 6 g j. 6 − Since p = reg ( ), and since f is ﬂat (this can be deduced, for example, from A OY the corollary of [30, Theorem 23.1, p. 179]) using Proposition 4.10 we deduce that there exists an exact complex

0 f ∗(F Ln) α f ∗(A−pg−m Ln) α . . . → g+1 ⊗ ⊗ → ⊕ ⊗ ⊗ → f ∗(A−m Ln) α f ∗(F Ln) α 0 → ⊕ ⊗ ⊗ → ⊗ ⊗ →

of sheaves on X. Using this complex, (4.10), and Lemma 4.11 we conclude that

Hj(X, f ∗(F Ln) α)=0, ⊗ ⊗

when j > q.

Our ﬁrst corollary generalizes Corollary 4.6 and is used in the proof of Theorem

4.3.

Corollary 4.12. Let X be an abelian variety and let E be vector bundle on X. Let L be

a non-degenerate line bundle on X. There exists an n0 > 0 with the property that, for all n > n , and all α Pic0(X), 0 ∈

Hj(X, Ln E α)=0 for j = i(L), and Hi(L)(X, Ln E α) =0. ⊗ ⊗ 6 ⊗ ⊗ 6

In particular, for every line bundle M on X, there exists an n such that Ln M is non- 0 ⊗ degenerate and i(Ln M)=i(Ln) for all n > n . ⊗ 0

Proof. Let q := i(L). Then i(L−1)= g q. If α Pic0(X), then by Theorem 4.2, there − ∈ ′ exists positive integers n0 and n0 such that

Hi(X, E Ln α)=0, for all n > n and all i > q, ⊗ ⊗ 0 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 47

and

′ Hi(X, E∨ L−n α−1)=0, for all n′ > n′ and all i>g q. ⊗ ⊗ 0 − Thus, by Serre duality,

Hi(X, E Ln α)=0 for all i = q and all n > max n , n′ . ⊗ ⊗ 6 { 0 0}

It remains to show that there exists an n such that Hq(X, E Ln α) =0 for all 0 ⊗ ⊗ 6 n > n and all α Pic0(X). By the Hirzebruch-Riemann-Roch Theorem, we have 0 ∈ (rank E) ng−1 χ(E Ln α)= ng c (L)g + c (E) c (L)g−1 + . . . . (4.11) ⊗ ⊗ g! 1 (g 1)! 1 1 ZX − g Since L is non-degenerate X c1(L) is nonzero. As a consequence, the right hand side of (4.11) is nonzero forR all n 0. ≫

A special case of Theorem 4.2 can be phrased in the language of q-ampleness in the sense of B. Totaro [53]. More precisely, a line bundle L on a projective variety X is said to be q-ample if, for all coherent sheaves F on X, there exists an n such that Hi(X, Ln F )=0, for 0 ⊗

all i > q and all n > n0.

Corollary 4.13. A non-degenerate line bundle on an abelian variety is q-ample if and only if its index is less than or equal to q.

Proof. Let L be a non-degenerate q-ample line bundle. Since L is q-ample, we conclude that Hj(X, Ln)=0 for j > q and all sufﬁciently positive integers n. Conse- quently, i(Ln) 6 q. Since i(L) = i(Ln), we conclude that i(L) 6 q. The converse is an immediate consequence of Theorem 4.2, taking Y = X and f the identity map. 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 48

Remark. If X is a simple abelian variety, then every degenerate line bundle, that is a line bundle L for which χ(L)=0, is an element of Pic0(X). We conclude that, for simple abelian varieties, if q

4.5.3 Proof and discussion of Theorem 4.1

Theorem 4.1 is a consequence of the results of 4.5.2 and the following proposition. § Proposition 4.14. Let L be a non-degenerate line bundle on X. If E and F are vector bundles on X then there exists an n such that T ∗(E) Ln α F satisﬁes the IC and 0 x ⊗ ⊗ ⊗ has index equal to that of L, for all x X, for all n > n , and all α Pic0(X). ∈ 0 ∈

Proof. Let q := i(L) and let α Pic0(X). The vector bundles m∗E p∗(Ln α F ), ∈ ⊗ 2 ⊗ ⊗ for n > 1, on X X are ﬂat over X, via p . Also, if x X, then × 1 ∈

m∗(E) p∗(Ln α F ) = T ∗(E) Ln α F . ⊗ 2 ⊗ ⊗ |{x}×X x ⊗ ⊗ ⊗

Since the Euler characteristic of a ﬂat family of sheaves over a connected base is constant, using Corollary 4.12 with x =0, we deduce that there exists an m0 > 0 with the property that for all n > m and all x X, 0 ∈

χ(T ∗(E) Ln α F ) =0. x ⊗ ⊗ ⊗ 6

As a result, to prove Proposition 4.14, it sufﬁces to establish the existence of a p such that if i = q, x X, and n > p then 0 6 ∈ 0

Hi(X, T ∗(E) Ln α F )=0. x ⊗ ⊗ ⊗ 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 49

Let A be a globally generated ample line bundle on X. Let m := regA(E),

′ ∨ m := regA(E ), and p := g +1. Using Corollary 4.12, we see that there exists an n0 such that Hℓ+j(X, Ln γ F A−pℓ−m)=0, (4.12) ⊗ ⊗ ⊗ for all j > q, for all 0 6 ℓ 6 g j, for all γ Pic0(X), and all n > n . Again, using − ∈ 0 ′ Corollary 4.12, we see that there exists an n0 such that

′ Hℓ+j(X, L−n γ∨ F ∨ A−pℓ−m )=0, (4.13) ⊗ ⊗ ⊗ for all j>g q, for all 0 6 ℓ 6 g j, for all γ Pic0(X), and all n > n′ . − − ∈ 0 Using Proposition 4.10, we obtain exact complexes

0 T ∗(E ) T ∗(A−gp−m) T ∗(A−m) T ∗(E) 0 (4.14) → x g+1 → ⊕ x →···→⊕ x → x → and

′ ′ ′ 0 T ∗(E ) T ∗(A−gp−m ) T ∗(A−m ) T ∗(E∨) 0 (4.15) → x g+1 → ⊕ x →···→⊕ x → x → of sheaves on X. Since T ∗A = A β, where β Pic0(X), using (4.12), (4.13), (4.14), (4.15), Serre x ⊗ ∈ duality, and Lemma 4.11, we conclude that

Hj(X, T ∗(E) Ln α F )=0, x ⊗ ⊗ ⊗ for all j = q, for all n > max n , n′ , for all α Pic0(X), and all x X. 6 { 0 0} ∈ ∈

Proof of Theorem 4.1. If x X and n > 1 then ∈

T ∗(Ln E)= Ln α T ∗(E) x ⊗ ⊗ ⊗ x 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 50

and T ∗(Ln E) M n F = Ln α T ∗(E) M n F , x ⊗ ⊗ ⊗ ⊗ ⊗ x ⊗ ⊗ where α is some element of Pic0(X).

Thus, by repeated application of Proposition 4.14, there exists an n0 such that the vector bundles

T ∗(Ln E), T ∗(Ln E) M n F , and M n F x ⊗ x ⊗ ⊗ ⊗ ⊗ satisfy the IC and have index, respectively,

i(L), i(L M), and i(M), ⊗

for all x X and all n > n . Theorem 4.1 now follows because ∈ 0

i(L M)=i(L)+i(M). ⊗

4.5.4 Proof and discussion of Theorem 4.3

Proposition 4.15 below, in conjunction with Corollary 4.9 (b), yields one proof of

Theorem 4.3.

Proposition 4.15. Let X be an abelian variety and let E and F be vector bundles on X. Let (L, M) be a pair of line bundles on X satisfying the PIC. Then, under this hypothesis,

there exists a positive integer n0 such that the cup-product

∪ Hi(L)(X, T ∗(Ln E)) Hi(M)(X, M n F ) Hi(L⊗M)(X, T ∗(Ln E) M n F ) (4.16) x ⊗ ⊗ ⊗ −→ x ⊗ ⊗ ⊗

is nonzero and surjective for all n > n and all x X. 0 ∈ 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 51

Proof. Let q := i(L M). Throughout the proof, we ﬁx a globally generated ample ⊗ line bundle A on X.

We ﬁrst show that the cup-products are surjective for all sufﬁciently large n not depending on the points of X. Let denote the ideal sheaf of the diagonal I∆ ∆ X X. It is enough to exhibit a positive integer n such that ⊆ × 0

Hq+1(X X, T ∗(Ln E) ⊠ (M n F ))=0 × I∆ ⊗ x ⊗ ⊗ for all n > n and all x X. 0 ∈ Set B := A ⊠ A, m := reg ( ) and p := 2g +1. Since L ⊠ M is non-degenerate B I∆

and has index q there exists, by Theorem 4.2, an n0 > 0 such that

Hq+1+ℓ(X X, (T id )∗(E ⊠ F B−pℓ−m Ln ⊠ M n) α)=0, (4.17) × x × X ⊗ ⊗ ⊗

for all α Pic0(X X), for all x X, for all 0 6 ℓ 6 2g q 1, and all n > n . ∈ × ∈ − − 0 Fix such an n , let n > n , and let x X. Observe now that 0 0 ∈

B−pℓ−m T ∗(Ln E)⊠(M n F ) = (T id )∗(E⊠F B−pℓ−m Ln ⊠M n) β, (4.18) ⊗ x ⊗ ⊗ ∼ x× X ⊗ ⊗ ⊗

where β is some element of Pic0(X X). × By Proposition 4.10, there exists an exact complex of sheaves

0 I B−2gp−m B−m 0 (4.19) → 2g+1 → ⊕ →···→⊕ → I∆ →

on X X. Tensoring (4.19) with T ∗(Ln E) ⊠ (M n F ) and using (4.18), (4.17), × x ⊗ ⊗ and Lemma 4.11, we conclude that

Hj(X X, T ∗(Ln E) ⊠ (M n F ))=0, × I∆ ⊗ x ⊗ ⊗

for all j > q, for all n > n , and all x X. 0 ∈ 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 52

To see that the target space of the cup-products are nonzero, for all sufﬁciently large n not depending on the points of X, note that, by Proposition 4.14, there exists an n0 > 0 such that

Hq(X, T ∗(E Ln) (F M n)) =0, (4.20) x ⊗ ⊗ ⊗ 6

for all x X and all n > n . ∈ 0 Since the cup-products are surjective for all sufﬁciently large n, not depending on the points of X, we conclude, using (4.20), that they are nonzero and surjective for all sufﬁciently large n not depending on the points of X.

Proof of Theorem 4.3. By Proposition 4.15, there exists a positive integer n0 with the property that the cup-products (T ∗(Ln E), M n F ) are nonzero and surjective ∪ x ⊗ ⊗ for all x X and all n > n . By Corollary 4.9 (b), this is equivalent to the vector ∈ 0 bundle Ri(L⊗M)(m∗(Ln E) p∗(M n F )) p1∗ ⊗ ⊗ 2 ⊗

being globally generated for all n > n0.

4.5.5 Theorem 4.3 and the work of Pareschi-Popa

Let (E, F ) be a pair of vector bundles on X which satisﬁes the PIC. We use Pareschi- Popa’s theory of M-regularity, see [45] and [46], to obtain a sufﬁcient condition for

i(E⊗F ) ∗ ∗ the vector bundle Rp ∗ (m E p F ) to be globally generated. 1 ⊗ 2 As a consequence, by Corollary 4.9 (b), we also obtain a sufﬁcient condition for the cup-products (T ∗E, F ), for x X, to be nonzero and surjective. Combining ∪ x ∈ these results with Theorem 4.1 and Corollary 4.12, we obtain a second proof of

Theorem 4.3. 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 53

Proposition 4.16. Let (E, F ) be a pair of vector bundles on X which satisﬁes the PIC. Let A be an ample line bundle on X. If

p∗(A−1 α) m∗E p∗F 1 ⊗ ⊗ ⊗ 2 satisﬁes the IC and has index i(E F ), for all α Pic0(X), then ⊗ ∈ Ri(E⊗F )(m∗E p∗F ) p1∗ ⊗ 2 is globally generated and the cup-products (T ∗E, F ) are surjective for all x X. ∪ x ∈

∗ ∗ i(E⊗F ) Proof. Let := m E p F and let := Rp ∗ ( ). By [44, Theorem 2.1, p. 654] N ⊗ 2 E 1 N or [45, Theorem 2.4, p. 289] (see also [6, Theorem 14.5.2]) to prove that is globally E generated it sufﬁces to prove that

Hi(X, A−1 α)=0, E ⊗ ⊗ for all i> 0 and all α Pic0(X). ∈ Let α Pic0(X). The push-pull formula implies that ∈ Rj (p∗(A−1 α) )= A−1 α Rj ( ), p1∗ 1 ⊗ ⊗N ⊗ ⊗ p1∗ N for all j. Hence, we have

A−1 α when j = i(E F ) j ∗ −1 ⊗ ⊗E ⊗ Rp ∗ (p1(A α) )=  1 ⊗ ⊗N  0 when j = i(E F ). 6 ⊗  Using the Leray spectral sequence we obtain

Hℓ−i(E⊗F )(X, α A−1) = Hℓ(X X,p∗(A−1 α) ). (4.21) E ⊗ ⊗ × 1 ⊗ ⊗N By assumption the right hand side of (4.21) is zero except when ℓ = i(E F ) so the ⊗ higher cohomology groups of α A−1 vanish as desired. E ⊗ ⊗ 4. ABELIAN VARIETIES: INDEX CONDITIONS AND CUP-PRODUCTS 54

Corollary 4.17. Let (L, M) be a pair of line bundles on X satisfying the PIC. If A is an ample line bundle and if p∗A−1 m∗L p∗M is a non-degenerate line bundle with index 1 ⊗ ⊗ 2 i(L M) then ⊗ Ri(L⊗M)(m∗L p∗M) p1∗ ⊗ 2 is globally generated and the cup-products (T ∗L, M) are surjective for all x X. ∪ x ∈

Proof. If p∗A−1 m∗L p∗M is non-degenerate and if i(A−1 p∗L p∗M)=i(L M) 1 ⊗ ⊗ 2 ⊗ 1 ⊗ 2 ⊗ then the same is true for p∗(A−1 α) m∗L p∗M, for all α Pic0(X), because 1 ⊗ ⊗ ⊗ 2 ∈ p∗(α) is an element of Pic0(X X). 1 ×

Second proof of Theorem 4.3. By Theorem 4.1, there exists an n0 with the property that the pair (Ln E, F n F ) satisﬁes the PIC and i(T ∗(Ln E) F n F )=i(L M), ⊗ ⊗ x ⊗ ⊗ ⊗ ⊗ for all x X, and all n > n . ∈ 0 Let A be an ample line bundle on X. Since m∗L p∗M is a non-degenerate ⊗ 2 line bundle with index i(L M) on X X, by Corollary 4.12 and increasing n if ⊗ × 0 necessary, we conclude that the vector bundles

p∗(A−1 α) m∗(Ln E) p∗(M n F ), 1 ⊗ ⊗ ⊗ ⊗ 2 ⊗ for all n > n , and all α Pic0(X), satisfy the IC and have index i(L M). 0 ∈ ⊗ i(L⊗M) ∗ n n Using Proposition 4.16, we conclude that Rp ∗ (m (L E) (M F )) is 1 ⊗ ⊗ ⊗ globally generated for all n > n0. By Corollary 4.9 (b), this is equivalent to the condition that the cup-products (T ∗(Ln E), M n F ) are nonzero and surjective ∪ x ⊗ ⊗ for all x X and all n > n . ∈ 0 5

Examples of cup-product problems on abelian varieties

We illustrate the results of Chapter 4 by considering examples. For instance, we

show that non-trivial cup-product problems can result in a zero map. We also construct families of abelian varieties for which every possible instance of our main result is realized for each member of the family.

5.1 Cup-products are not, in general, nonzero

Let E be an elliptic curve and let X denote the product E E. We prove that X × admits a pair of line bundles (L, M) which satisﬁes the PIC and a curve C X for ⊆ which the cup-product (T ∗L, M) is zero, for all x C, and nonzero for all x C. ∪ x ∈ 6∈ On the other hand, note that a special case of Theorem 4.3 is that–in contrast to this phenomena–after scaling things behave more uniformly.

55 5. EXAMPLES OF CUP-PRODUCT PROBLEMS ON ABELIAN VARIETIES 56

5.1.1 The Neron-Severi space of X

Let x E, let f denote the numerical class of the divisor x E, and let f denote ∈ 1 { }× 2 the numerical class of the divisor E x . Finally let ∆ denote the numerical class ×{ } of the diagonal and let γ denote the numerical class f + f ∆. 1 2 − 1 1 Let N (X)R denote the real Neron-Severi space of X. Then dimR N (X)R > 3

and the classes f1, f2 and γ span a three dimensional subspace. The intersection

1 table and the subspace of N (X)R associated to the classes f1, f2 and γ is pictured below: If χ(L) 6= 0 and The intersection relations. L has numerical class af1 + bf2 + cγ then

· f1 f2 γ H 0 0 iff ab − c2 > 0 and a + b> 0 8 f1 0 1 0 > H1 i(L) = >1 iff ab − c2 < 0 2 > f2 1 0 0 ab − c = 0 < 2 H 2 >2 iff ab − c > 0 and a + b< 0. γ 0 0 −2 > :>

See also [20, Ex. V.1.6, p. 367].

5.1.2 Cup-products on X

Using 5.1.1, we see that the numerical classes mf + γ, for m > 3, and f f § − 1 1 − 2 determine cup-product maps

(L, M) : H1(X, L) H1(X, M) H2(X, L M) ∪ ⊗ → ⊗ for all L with numerical class mf + γ and all M with numerical class f f . − 1 1 − 2 If, for n > 1, the multiplication map (Ln, M n) is surjective then ∪

h1(X, Ln)h1(X, M n) > h2(X, Ln M n). ⊗ 5. EXAMPLES OF CUP-PRODUCT PROBLEMS ON ABELIAN VARIETIES 57

Observe that h1(X, Ln)= n2, h1(X, M n)= n2, and h2(Ln M n)= n2(m 2). Hence, ⊗ − if (Ln, M n) is surjective then n > √m 2. In particular, (L, M) is not surjective ∪ − ∪ for m > 4. We discuss the boundary case m =3 in 5.1.3. §

5.1.3 Cup-products with nonzero source and target space can re-

sult in a zero map

Fix line bundles L and M on X with numerical classes 3f + γ and f f , re- − 1 1 − 2 spectively, and consider the family of cup-product maps

(T ∗L, M) : H1(X, T ∗L) H1(X, M) H2(X, T ∗L M) (5.1) ∪ x x ⊗ → x ⊗ parametrized by points of X. Since the source and target space of these maps are

1-dimensional vector spaces each map is either zero or surjective. To determine the nature of these maps recall that, by Proposition 4.8, the image of (T ∗L, M), x X, coincides with the image of the evaluation map ∪ x ∈

H0(X, R2 (m∗L p∗M)) κ(x) R2 (m∗L p∗M) . (5.2) p1∗ ⊗ 2 ⊗ → p1∗ ⊗ 2 |x

Also R2 (m∗L p∗M) is a nontrivial line bundle and p1∗ ⊗ 2

h0(X, R2 (m∗L p∗M))=1. p1∗ ⊗ 2

(Apply Proposition 4.7 and Corollary 4.9 (c) or Proposition 4.7 and (5.3) below.) As a result if C is the base locus of R2 (m∗L p∗M), then the evaluation map (5.2) p1∗ ⊗ 2 is zero for all x C and is nonzero for all x C. Since the image of (5.2), for a ∈ 6∈ ﬁxed x X, coincides with that of (T ∗L, M) we conclude that (T ∗L, M) is zero ∈ ∪ x ∪ x if x C and nonzero if x C. ∈ 6∈ 5. EXAMPLES OF CUP-PRODUCT PROBLEMS ON ABELIAN VARIETIES 58

5.1.4 The ﬁrst Chern class

We can gain more precise information regarding the nature of the vector bundles considered in 5.1.3. § Let X be an abelian surface. Let K(X) and K(X X) denote the Grothendieck × group of coherent sheaves on X and X X respectively. Let L and M be line × bundles on X. Let y denote the class of m∗L p∗M in K(X X) and let ⊗ 2 ×

x := p (y) K(X). 1∗ ∈

1 Let ch1(x) denote the portion of ch(x) contained in A (X)Q, where

∗ ch : K(X) A (X)Q →

is the Chern character homomorphism. We now prove that

1 ch (x)= χ(M) c (L)+ χ(L) c (M) A (X)Q. (5.3) 1 1 1 ∈

By the Grothendieck-Riemann-Roch theorem, we have

1 ch (x)= p (c (m∗L p∗M)3). (5.4) 1 3! 1∗ 1 ⊗ 2

On the other hand, expanding and noting that Chern classes commute with ﬂat pull-back, we have

c (m∗L p∗M)3 = 3(m∗ c (L)2p∗ c (M)+ m∗ c (L)p∗ c (M)2). 1 ⊗ 2 1 1 1 1 1 1

Consequently,

p (c (m∗L p∗M)3)=3 c (L)2 c (M)+ c (M)2 c (L) . (5.5) 1∗ 1 ⊗ 2 1 1 1 1 ZX ZX Combining (5.4) and (5.5) we conclude that (5.3) holds. 5. EXAMPLES OF CUP-PRODUCT PROBLEMS ON ABELIAN VARIETIES 59

5.2 The pair index condition and number ﬁelds

Let K denote a totally real number ﬁeld of degree g over Q, let σ1,...,σg denote the embeddings of K into R, and let denote the ring of integers of K. OK A complex torus X admits multiplication by K if there exists an embedding

.K ֒ End(X) Z Q → ⊗

Let h denote the upper half plane, let z =(z ,...,z ) hg, and let 1 g ∈

g Λz := (z σ (α)+ σ (β),...,z σ (α)+ σ (β)) : α, β C . { 1 1 1 g g g ∈ OK } ⊆

g g Then Λz is the Z-linear span of an R-basis for C ; the quotient Xz := C /Λz is a complex torus and has multiplication by K, [6, 9.2]. § Theorem 5.1. Let p and q be nonnegative integers whose sum is less than or equal to g.

g Let z = (z ,...,z ) h . The complex torus Xz admits line bundles L and M such that 1 g ∈ L, M, and L M are non-degenerate, i(L)= p, i(M)= q, and i(L M)= p + q. ⊗ ⊗

Proof. First, note that every η determines a Hermitian form ∈ OK g σ (η) H : Cg Cg C deﬁned by H (x, y) := i x y η × → η Im z i i i=1 i X for x =(x ,...,x ) and y =(y ,...,y ) Cg. 1 g 1 g ∈ Also if x, y Λz and ∈

x =(z1σ1(α)+ σ1(β),...,zgσg(α)+ σg(β)),

y =(z1σ1(γ)+ σ1(ζ),...,zgσg(γ)+ σg(ζ)),

with α, β, γ, and ζ , then ∈ OK

Im H (x, y) = Tr Q(ηαζ) Tr Q(ηβγ). η K/ − K/ 5. EXAMPLES OF CUP-PRODUCT PROBLEMS ON ABELIAN VARIETIES 60

Thus, the imaginary part of Hη is integral on Λz. Fix disjoint subsets I,J 1,...,g of cardinalities p and q respectively. Let ⊆ { } U R2g be the subset consisting of those points (x ,...,x ,y ,...,y ) R2g such ⊆ 1 g 1 g ∈ that x < 0 if k I, x > 0 if k I, y < 0 if k J, y > 0 if k J, and x + y < 0 k ∈ k 6∈ k ∈ k 6∈ k x for all k I J. Then U is open in R2g. On the other hand, the set ∈ ∪

S := (σ (η),...,σ (η), σ (β),...,σ (β)):(η, β) K2 { 1 g 1 g ∈ } is dense in R2g. (This follows for example from [29, p. 135].) We conclude that the intersection of S with U is nonempty.

Considering the deﬁnitions of U and S, we conclude that (after scaling by a positive integer if necessary) there exist η, β with the property that ∈ OK (a) σ (η) < 0 if k I and σ (η) > 0 if k I; k ∈ k 6∈ (b) σ (β) < 0 if k J and σ (β) > 0 if k J; k ∈ k 6∈ (c) σ (η)+ σ (β) < 0 if k I J. k k ∈ ∪

Consequently, the Hermitian forms Hη, Hβ, and Hη +Hβ = Hη+β have, respectively, exactly p, q, and p + q negative eigenvalues.

Let χHη and χHβ be semi-characters for Λ with respect to Hη and Hβ respectively.

The line bundles L(Hη, χHη ), L(Hβ, χHβ ), L(Hη + Hβ, χHη χHβ ) are non-degenerate, see [41, p. 80], and satisfy the relation

i(L(Hη, χHη ))+i(L(Hβ, χHβ )) = i(L(Hη + Hβ, χHη χHβ )) by the corollary on p. 151 of [41].

g Remark. Shimura proved that there exist z h for which EndQ(Xz) = K, see ∈ [51, Theorem 5, p. 176] or [6, Theorem 9.9.1, p. 274]. Using this result, together

with the fact that K contains no nontrivial idempotents, and [6, Theorem 5.3.2 p. 5. EXAMPLES OF CUP-PRODUCT PROBLEMS ON ABELIAN VARIETIES 61

123] we conclude that there exists simple abelian varieties for which Proposition 5.1 applies.

5.3 The classical case: theta groups and cup-products

We now describe an approach, used by Mumford, to study of cup-product problems arising from pairs of algebraically equivalent ample line bundles. In 5.3.3 § we consider this technique as it applies to non-degenerate line bundles having nonzero index.

5.3.1 Preliminaries

Let X be an abelian variety. Every line bundle L on X determines a group homomorphism φ : X Pic0(X) deﬁned by x T ∗L L−1. See [41, The theorem of L → 7→ x ⊗ the square, p. 57] and [41, 8, p. 70]. § Let K(L) := x X : T ∗L = L , and observe that x K(L) if and only if { ∈ x ∼ } ∈ x ker φ . Furthermore, if L and M are line bundles on X and x K(L M), then ∈ L ∈ ⊗ −1 φL(x)= φM (x) . ∼ Let G(L M) := (x, φ) : φ : L = T ∗L denote the theta group of the line ⊗ { −→ x } bundle L M, see [36, p. 289] and [39, p. 64]. The group G(L M) acts on the ⊗ ⊗ cohomology groups Hi(X, L M). Explicitly, we have ⊗

(x, φ) σ := Hi(φ−1) Hi(T ∗)(σ), for σ Hi(X, L M) and (x, φ) G(L M). · ◦ x ∈ ⊗ ∈ ⊗

See [39, p. 66]. 5. EXAMPLES OF CUP-PRODUCT PROBLEMS ON ABELIAN VARIETIES 62

5.3.2 Cup-products, ample line bundles, and theta groups

We now assume that X is deﬁned over the complex numbers. One approach to study cup-product maps

(Ln, M m) : H0(X, Ln) H0(X, M m) H0(X, Ln M m) ∪ ⊗ → ⊗ arising from pairs of algebraically equivalent ample line bundles L and M is to consider the natural map

H0(X, L φ (x) α) H0(X, M φ (x) α−1) H0(X, L M) (5.6) ⊗ L ⊗ ⊗ ⊗ M ⊗ → ⊗ x∈K(ML⊗M) arising from elements α of Pic0(X) [39, 3, p. 62–70]. § The lemma on [39, p. 68] implies that the image of the map (5.6) is (nonzero and) stable under the action of G(L M). Since H0(X, L M) is an irreducible ⊗ ⊗ G(L M)-module the map (5.6) is surjective. ⊗ Using the surjectivity of the map (5.6), Mumford proved that if n, m > 4 then

(Ln, M m) is surjective [39, Theorem 9, p. 68]. This statement was latter improved. ∪ For example, S. Koizumi proved that if n > 2 and m > 3 then (Ln, M m) is surjec- ∪ tive [24, Theorem 4.6, p. 882].

5.3.3 Cup-products, theta groups, and line bundles with nonzero

index

Compare the situation of 5.3.2 with that of 5.1. Speciﬁcally, 5.1.2 shows that we § § § cannot expect to obtain general results to the effect that (Ln, M m) is surjective ∪ for speciﬁc positive integers n and m independent of the pair (L, M) satisfying the

PIC. 5. EXAMPLES OF CUP-PRODUCT PROBLEMS ON ABELIAN VARIETIES 63

Also note that if (L, M) is a pair of line bundles on X which satisﬁes the PIC and if both i(L) and i(M) are positive then they cannot be algebraically equivalent.

In spite of these issues, we can still use some aspects of the approach described in 5.3.2 to gain insight into the nature of cup-product problems arising from non- § degenerate line bundles with nonzero index. For example, ﬁx a pair (L, M) of line bundles on X which satisﬁes the PIC. If α is an element of Pic0(X), if x K(L M), andif β := α φ (x)= α φ (x)−1 then, by ∈ ⊗ ⊗ L ⊗ M choosing isomorphisms, φ : L β T ∗(L α) and ψ : M β−1 T ∗(M α−1), ⊗ → x ⊗ ⊗ → x ⊗ we obtain an isomorphism

φ ψ : L M T ∗(L α) T ∗(M α−1)= T ∗(L M). ⊗ ⊗ → x ⊗ ⊗ x ⊗ x ⊗

Using this isomorphism we obtain a commutative diagram

∪(L⊗β,M⊗β−1) Hi(L)(X, L β) Hi(M)(X, M β−1) / Hi(L⊗M)(X, L M) ⊗ ⊗ ⊗ ⊗ Hi(L)(φ)⊗Hi(M)(ψ) Hi(L⊗M)(φ⊗ψ) Hi(L)(X, T ∗(L α)) Hi(M)(X, T ∗(M α−1)) Hi(L⊗M)(X, T ∗(L M)) x ⊗ ⊗ x ⊗ x ⊗ i(L) ∗ i(M) ∗ i(L⊗M) ∗ H (T−x)⊗H (T−x) H (T−x) ∪(L⊗α,M⊗α−1) Hi(L)(X, L α) Hi(M)(X, M α−1) / Hi(L⊗M)(X, L M) ⊗ ⊗ ⊗ ⊗ from which we deduce

dim image (L α, M α−1) = dim image (L φ (x) α, M φ (x) α−1). ∪ ⊗ ⊗ ∪ ⊗ L ⊗ ⊗ M ⊗

In other words, the dimension of the image of the map (L α, M α−1) is the ∪ ⊗ ⊗ same for all elements of Pic0(X) in the orbit of α under the action of K(L M) on ⊗ Pic0(X) deﬁned by x α := φ (x) α. · L ⊗ Moreover, by using the fact that Hi(L⊗M)(X, L M) is an irreducible G(L M)- ⊗ ⊗ module, we can deduce, in a manner similar to 5.3.2, that if the cup-product map § 5. EXAMPLES OF CUP-PRODUCT PROBLEMS ON ABELIAN VARIETIES 64

(L α, M α−1) is nonzero then the natural map ∪ ⊗ ⊗

Hi(L)(X, L φ (x) α) Hi(M)(X, M φ (x) α−1) Hi(L⊗M)(X, L M) ⊗ L ⊗ ⊗ ⊗ M ⊗ → ⊗ x∈K(ML⊗M) (5.7) is nonzero and surjective. Note, 5.1.3 shows that the map (5.7) can be the zero § map. On the other hand, by Corollary 4.9, there exists α Pic0(X) for which the ∈ cup-product map (L α, M α−1) is nonzero whence the map (5.7) is surjective. ∪ ⊗ ⊗ In more detail, Corollary 4.9 (a) implies that there exists an x X with the ∈ property that (T ∗L, M) is nonzero. Since L M is non-degenerate there exists ∪ x ⊗ y X for which T ∗(T ∗L M) = L M and, hence, there exists a commutative ∈ y x ⊗ ⊗ diagram

∪(T ∗L,M) Hi(L)(X, T ∗L) Hi(M)(X, M) x / Hi(L⊗M)(X, T ∗L M) x ⊗ x ⊗ i(L) ∗ i(M) ∗ i(L⊗M) ∗ H (Ty )⊗H (Ty ) H (Ty ) ∪(L⊗α,M⊗α−1) Hi(L)(X, L α) Hi(M)(X, M α−1) / Hi(L⊗M)(X, L M) ⊗ ⊗ ⊗ ⊗ where α is an appropriate element of Pic0(X). Since the top horizontal arrow is nonzero the same is true for the bottom horizontal arrow. Hence (L α, M α−1) ∪ ⊗ ⊗ is nonzero so that the map (5.7) is surjective. 6

Theta groups and vector bundles on abelian varieties

Let X be an abelian variety deﬁned over an algebraically closed ﬁeld k. We con-

sider theta groups associated to simple semi-homogenous vector bundles of separable type on X. We determine the structure and representation theory of these groups. In doing so, we relate work of Mumford [36], Mukai [33], and Umemura [55].

6.1 Formulation and statement of results

If x X, then let T : X X denote translation by x in the group law. ∈ x → Deﬁnitions. Let E be a vector bundle on X. E is non-degenerate if χ(E) =0; • 6 E is simple if End (E)= k; • OX E is semi-homogeneous if for all x X there exists a line bundle L on X with • ∈ the property that T ∗E = E L; x ∼ ⊗ 65 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 66

E is of separable type if E is non-degenerate and if χ(E) is not divisible by the • characteristic of k.

Let E be a simple semi-homogeneous vector bundle of separable type on X. If x X, then let Aut (E) denote the set of isomorphisms E T ∗E of -modules. ∈ x → x OX Mukai has shown that χ(E)2 = #K(E) where K(E) is the (ﬁnite group) deﬁned by

K(E) := x X(k) : Aut (E) = ∅ . { ∈ x 6 }

See [33, 6, 7, and Corollary 7.9, p. 271] for example. § § The theta group of E, which we denote by G(E) and deﬁne in 6.2.2, is a central § extension of k× by K(E).

Our ﬁrst result generalizes [36, Theorem 1] and sheds light on a problem posed by Umemura [55, p. 120]:

Theorem 6.1. Let E be a simple semi-homogenous vector bundle of separable type on X. The theta group G(E) is a non-degenerate central extension of k× by K(E).

A consequence of Theorem 6.1 is that every simple semi-homogenous vector

bundle of separable type E on X determines a sequence of integers d =(d1,...,dp), d d , which we refer to as the type of its theta group G(E). Using Theorem 6.1, i | i+1 we determine the representation theory of G(E).

Theorem 6.2. Let E be a simple semi-homogenous vector bundle of separable type on X.

Let d =(d1,...,dp) be the type of G(E). We have the following (a) the theta group G(E) admits exactly gcd(n, d )2 gcd(n, d )2 non-isomorphic ir- 1 ··· p reducible weight n G(E)-module(s);

(b) aweight n representation is irreducible if and only if it has dimension d1···dp ; gcd(n,d1)··· gcd(n,dp) 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 67

(c) every weight n G(E)-module decomposes into a direct sum of irreducible weight n G(E)-modules. Every G(E)-module decomposes into a direct sum of weight n

G(E)-modules.

We also prove that simple semi-homogeneous vector bundles of separable type admit exactly one nonzero cohomology group. (Proposition 6.17.) Thus, as a consequence of Theorems 6.1 and 6.2, we obtain the following corollary.

Corollary 6.3. Let E be a simple semi-homogenous vector bundle of separable type on

X. The unique nonzero cohomology group Hi(E)(X, E) is the unique irreducible weight 1 representation of G(E).

We prove Theorem 6.1 in 6.3.2. We prove Theorem 6.2 and Corollary 6.3 in § 6.4.3. §

6.2 Construction of theta groups

Let X be an abelian variety deﬁned over an algebraically closed ﬁeld k. We generalize those parts of [36, 1] which we need in 6.3.2 to prove Theorem 6.1. § §

6.2.1 Preliminaries from descent theory

Let K X be a ﬁnite subgroup. We assume that the order of K is not divisible by ⊆ the characteristic of k. We consider descent of quasi-coherent sheaves with respect

to the quotient map f : X Y = X/K. → Let F be a quasi-coherent sheaf on X. If x X(k), then let ∈

Aut (F ) := φ Hom (F, T ∗F ) : φ is an isomorphism . x { ∈ OX x } 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 68

If Aut (F ) = ∅, then it is an Aut(F )-torsor (or principal homogeneous space); that x 6 is to say

the group Aut(F ) acts on Aut (F ), for all x X; • x ∈ if x X and φ Aut (F ) then the group α Aut(F ) : α φ = φ is trivial; • ∈ ∈ x { ∈ · } if x X then Aut (F ) = α φ : α Aut(F ) for some (and hence every) • ∈ x { · ∈ }

element φ of Autx(F ). More succinctly, Aut(F ) acts simply transitively on Aut (F ), for all x X, when- x ∈ ever Aut (F ) = ∅. x 6 Covering datum for F , with respect to f,isapair (F,φ) where φ is a set consisting of exactly one φ Aut (F ), for all x K. Clearly if F admits covering datum then x ∈ x ∈ Aut (F ) = ∅ for all x K. x 6 ∈ Covering datum (F,φ) is said to be descent datum if the diagram

∗ φx ∗ Tx φy ∗ ∗ F UU / T F / T (T F ) (6.1) UUUU x x y UUU φx+y UUUU φy UUUU UUUU T ∗φ UUU ∗ y x ∗ ∗ * ∗ Ty F / Ty (Tx F ) Tx+yF commutes for all x, y K. ∈ Let Descent(QCoh(X), f) denote the category whose objects are descent data (F,φ) and where a morphism between pairs (F,φ) and (G, ψ) is given by an ele-

ment α of HomOX (F,G) having the property that the diagram

α F / G

φx ψx ∗ ∗ Tx α ∗ Tx F / Tx G

commutes for all x K. ∈ 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 69

Let H be a quasi-coherent sheaf on Y . If x K, then let can (H) be the isomor- ∈ x phism

f ∗H =(f T )∗H = T ∗(f ∗H), ◦ x x and let can(H) := can (H) : x K . { x ∈ } The pair (f ∗H, can(H)) is an object of Descent(QCoh(X), f).

We say that descent data (F,φ) is effective if

(F,φ) ∼= (H, can(H))

for some H QCoh(Y ). In this situation, we also say that F descends, via f, to H. ∈ The maps

H f ∗H, φ Hom (H,J) f ∗φ Hom((f ∗H, can(H)), (f ∗J, can(J))) 7→ ∈ OY 7→ ∈

determine a functor f ∗ : QCoh(Y ) Descent(QCoh(X), f). Grothendieck has proven → that this functor is an equivalence of categories. See [7, Theorem 4, p. 134], for in-

stance, as well as [41, Proposition 2, p. 66 and Theorem 1, p. 104].

Lemma 6.4. If a coherent sheaf E on X descends, via a separable isogeny f : X Y , to → a coherent sheaf F on Y , then

dimk EndOX (E) > dimk EndOY (F ).

Proof. Since the descent functor is fully faithful the k-vector space EndOY (F ) is

identiﬁed with the subspace of EndOX (E) which commutes with the descent data. 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 70

6.2.2 Theta groups and quasi-coherent sheaves

Let F be a quasi-coherent sheaf on X. Let K(F ) := x X(k) : Aut (F ) = ∅ . Let { ∈ x 6 } G(F ) := (x, φ) : x K(F ) and φ Aut (F ) . { ∈ ∈ x } If (x, φ) and (y, ψ) are elements of G(F ), then (x, φ) (y, ψ):=(x + y,φ ψ), where · ∗ φ ψ is the element of Aut (F ) determined by the composition ∗ x+y ∗ ψ Ty φ F T ∗F T ∗(T ∗F )= T ∗ (F ). −→ y −−→ y x x+y The pair (G(F ), ) is a group and the homorphisms: · ι π Aut(F ) F G(F ), deﬁned by α (0, α), and G(F ) F K(F ), deﬁned by (x, φ) x, −→ 7→ −→ 7→ determine a short exact sequence of groups

ι π 1 Aut(F ) F G(F ) F K(F ) 0. → −→ −→ →

6.2.3 Level subgroups and descent

Let F be a quasi-coherent sheaf on X. Deﬁnitons.

A subgroup K G(F ) is a level subgroup if it is ﬁnite, has order not divisible • ⊆ by the characteristic of k, and if the homomorphism π K : K π (K) is e F | → F injective. e e e Let K K(F ) and let K G(F ) be a level subgroup. We say that K lies over • ⊆ ⊆ K if πF (K)= K. e e

Proposition 6.5.e Let F be a quasi-coherent sheaf on X and let K K(F ) be a subgroup. ⊆ There exists a one to one correspondence between level subgroups K G(F ) lying over ⊆ K and (effective) descent datum (F,φ) with respect to the quotient map f : X X/K. e → 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 71

Proof. Let σ be the inverse of π K. If x K, then let φ be the element of Aut (F ) F | ∈ x x determined by σ. Let φ be the sete consisting of these isomorphisms. Since σ is a group homomorphism, the diagram (6.1) commutes for all x, y K. Hence, the ∈ pair (F,φ) is descent datum. Conversely, if (F,φ) is descent datum, then deﬁne

K := (x, φ ) : φ φ . { x x ∈ } e The fact that the diagram (6.1) commutes, for all x, y K, implies that K is a ∈ subgroup. In addition, the map (x, φ ) x is an isomorphisms of groups. x 7→ e It is clear, by construction, that the correspondences just deﬁned are mutual

inverses.

6.2.4 Theta groups and isogenies

Let F be a quasi-coherent sheaf on X, K G(F ) a level subgroup, and (F,φ) the ⊆ descent datum determined by K. Let K := π (K), Y := X/K, and f : X Y the e F → quotient map. e e Since (F,φ) is effective there exists a quasi-coherent sheaf G on Y with the

∗ property that (F,φ) ∼= (f G, can(G)). In particular, there exists an isomorphism α : f ∗G F of -modules. → OX

We now use α to relate G(G), G(F ), and the centralizer CK(G(F )) of K in G(F ). First, observe that every x X determines a morphism e ∈ e

f ∗ T ∗(α)◦?◦α−1 Aut (G) Aut (f ∗G) x Aut (F ) (6.2) f(x) −→ x −−−−−−−→ x

of Aut(G)-sets. In particular, we have f −1(K(G)) K(F ). ⊆ 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 72

Also if x and y are elements of X then the diagram

∗ −1 ∗ −1 ∗× ∗ T (α)◦?◦α ×T (α)◦?◦α f f ∗ ∗ y x Autf(y)(G) × Autf(x)(G) / Auty(f G) × Autx(f G) / Auty(F ) × Autx(F )

∗ ∗ ∗ ∗ − ∗ T (α)◦?◦α 1 f ∗ x+y Autf(x)+f(y)(G) / Autx+y(f G) / Autx+y(F ) (6.3) of Aut(G)-sets commutes.

Proposition 6.6. Let F be a quasi-coherent sheaf on an abelian variety X. Let K G(F ) ⊆ be a level subgroup, and let (F,φ) be the descent datum determined by K. Ine addition, let K := π (K), let Y := X/K, and let f : X Y be the quotient map. Let G be a F → e ∗ quasi-coherente sheaf on Y with the property that (F,φ) ∼= (f G, can(G)). Then:

(a) if CK (G(F )) denotes the centralizer of K in G(F ), then CK(G(F )) coincides with e e the set e

(x, η) : f(x) K(G) and η = T ∗(α) f ∗ψ α−1 for some ψ Aut (G) ; { ∈ x ◦ ◦ ∈ f(x) }

(b) themap C (G(F )) G(G) deﬁned by (x, η) (f(x), ψ), where ψ is the (unique) K → 7→ e element of Autf(x)(G) whose pullback is η, is a surjective group homomorphism with kernel equal to K.

Proof. We ﬁrst provee (a). If w K, then let φ Aut (F ) be the unique isomor- ∈ w ∈ w phism F T ∗F having the property that (w,φ ) K. → w w ∈ We know that (F,φ) descends to G and that if x X and y = f(x), then e ∈ ∗ ∗ ∗ (Tx F, Tx φ) descends to Ty G. The centralizer of K in G(F ) consists exactly of those (x, ψ) G(F ) with the ∈ property that ψ φ = φ ψ for all w K. Considering this condition, we conclude ∗ w ew ∗ ∈ that C (G(F )) consists exactly of those (x, ψ) G(F ) such that ψ determines an K ∈ isomorphisme of the pairs (F,φ) (T ∗F, T ∗φ). → x x 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 73

Thus, to determine C (G(F )), we need to examine, for a ﬁxed x K(F ), those K ∈ ψ isomorphisms F T ∗F ewhich commute with the descent data. −→ x Let x X and let y = f(x). The map ∈

Hom (G, T ∗G) Hom((F,φ), (T ∗F, T ∗φ)) (6.4) OY y → x x is given by η T ∗(α) f ∗η α−1 and is an isomorphism (since the descent functor 7→ x ◦ ◦ is fully faithful). Furthermore, under the map (6.4) isomorphisms carry over to isomorphisms. ψ In particular, if F T ∗F is an isomorphism which commutes with descent data −→ x then f(x) K(G). Considering the discussion above, we conclude that (a) holds. ∈ To prove (b) using (6.3), we check that the map is a group homomorphism. To see that the map is surjective, let y K(G). Then y = f(x) for some x K(F ). ∈ ∈

Since the map (6.4) is an isomorphism, every element of Auty(G) is in the image. Using the deﬁnition of the map, we check that its kernel is K.

Remark. Using the fact that K is a level subgroup of G(F ),e we can check that the

centralizer of K in G(F ) equalse its normalizer. e 6.3 Non-degenerate theta groups

Let X be an abelian variety deﬁned over an algebraically closed ﬁeld k. Let E be a simple semi-homogeneous vector bundle of separable type on X. In 6.3.2, § we prove that its theta group G(E) is a non-degenerate central extension of k× by

K(E). 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 74

6.3.1 Preliminaries on central extensions of k× by a ﬁnite abelian

group

Let K be a ﬁnite abelian group and assume that the order of K is not divisible by the characteristic of k. We consider a central extension

ι π 1 k× G K 0 (6.5) → −→ −→ →

of k× by K. Deﬁnitions.

The extension (6.5) is non-degenerate if ι(k×) equals the center of G. • A subgroup K G is a level subgroup if the homomorphism π K : K π(K) • ⊆ | → is injective. e e e e The extension (6.5) determines a bilinear form

[ , ] :K K k×, deﬁned by [x, y] := aba−1b−1, (6.6) − − G × → G

where a and b are any elements of G lying over x and y respectively.

−1 The form (6.6) is skew symmetric, that is, [x, x]G = 1, hence [x, y]G = [y, x]G , for all x, y G. If, in addition, the extension (6.5) is non-degenerate, then form ∈ (6.6) is also non-degenerate, that is for all 0 = x K there exists y K such that 6 ∈ ∈ [x, y] =1. G 6 We determine the structure of non-degenerate central extensions of the form (6.5). (See Proposition 6.9.) To achieve this, we ﬁrst prove two auxiliary results. (Proposition 6.7 and Lemma 6.8.)

Proposition 6.7. If [ , ]:K K k× is a non-degenerate skew symmetric bilinear − − × → form, then K admits subgroups K1 and K2 with the properties that: 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 75

(a) K= K K , [x ,y ]=1 for all x ,y K , i =1, 2; 1 ⊕ 2 i i i i ∈ i (b) the bilinear form , : K K k×, deﬁned by (x , x ) [x , x ], is non- h− −i 1 × 2 → 1 2 7→ 1 2 degenerate.

Proof. We prove (a) and (b) simultaneously by inducting on the number of elementary divisors of K. The base case is when K has no elementary divisors in which case statements (a) and (b) hold. Suppose now that d1 is the largest elementary divisor of K. Let x K be an element of order d . Since d is the largest elementary 1 ∈ 1 1 divisor of K, and since [ , ] is non-degenerate and skew-symmetric, there exists − − y K, of order d and not contained in x , with the property that [x ,y ] = ζ , 1 ∈ 1 h 1i 1 1 d1 × where ζd1 is a primitive d1th root of unity in k . Let W be the subgroup of K generated by x1 and y1 and let

W ⊥ := z K : [x , z]=1 and [y , z]=1 . { ∈ 1 1 }

Then W = x y and W W ⊥ = 1 . On the other hand, if z K then h 1i⊕h 1i ∩ h Ki ∈ n m ζd1 = [x1, z] and ζd1 = [y1, z] for some integers n and m. As a consequence, if w = mx + ny then w W , z = w + z w, and z w W ⊥. Hence, K= W W ⊥. 1 1 ∈ − − ∈ ⊕ Also the restriction of [ , ] to W ⊥ is a non-degenerate skew-symmetric bilinear − − form and W ⊥ is a proper subgroup of K. By induction, the statement holds for W ⊥ and we deduce that the statement holds for K.

The following lemma gives a sufﬁcient condition for producing level subgroups of G.

Lemma 6.8. Let G be a central extension of k× by K. If K K is a subgroup, having ⊆ the property that [x, y] = 1 for all x, y K, then G admits a level subgroup K with the G ∈ property that π(K)= K. e e 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 76

Proof. Write K as an internal direct sum K = n g and let d be the order of g . i=1h ii i i di Let hi be an element of G lying over gi, for all Li =1,...,n. Then hi corresponds to a unique α k×. Since k× is divisible there exists β k× such that βdi = α . Set i ∈ i ∈ i i z = h β−1. Then z is an element of G and has order d . Let K := z ,...,z . Then i i i i i h 1 ni K = n z and K is a subgroup of G having the property that K = π(K). Since i=1h ii e [x, y] L=1, for all x, y K, we deduce that π K : K K is an isomorphism. e G e ∈ | → e Our next proposition shows that every non-degeneratee e central extension of k× by K takes a more desirable form.

Proposition 6.9. If G is a non-degenerate central extension of k× by K then K admits subgroups K ,K K with the properties that 1 2 ⊆ (a) K= K K and [x ,y ] =1 if x ,y K , for i =1, 2; 1 ⊕ 2 i i G i i ∈ i (b) the bilinear form

, : K K k×, deﬁned by (x , x ) [x , x ] h− −i 1 × 2 → 1 2 7→ 1 2 G is non-degenerate;

(c) the extension G is equivalent to the extension G , where G := k× K K h−,−i h−,−i × 1 ⊕ 2 and where multiplication is deﬁned by

(α, x , x ) (β,y ,y )=(αβ x ,y , x + y , x + y ). 1 2 · 1 2 h 1 2i 1 1 2 2 Proof. Statements (a) and (b) are immediate consequences of Proposition 6.7. Let

K1 and K2 be level subgroups of G lying over K1 and K2 respectively. (Lemma 6.8

impliesf thatf such groups exist.) Fix group theoretic sections σ of π K , for i = 1, 2. If x = x + x K, with i | i 1 2 ∈ x K , then deﬁne i ∈ i f σ(x) := σ (x ) σ (x ). 2 2 · 1 1 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 77

This defines a normalized set-theoretic section σ :K G. → Let [ , ] denote the resulting factor set and let x = x + x and y = y + y , − − σ 1 2 1 2 x ,y K , be elements of K. To prove that G is equivalent to G , it suffices to i i ∈ i h−,−i check that x ,y = [x, y] . (6.7) h 1 2i σ By definition of , , we have h− −i

x ,y := [x ,y ] = σ (x ) σ (y ) σ (x )−1 σ (y )−1 (6.8) h 1 2i 1 2 G 1 1 · 2 2 · 1 1 · 2 2 while [x, y] := σ(x) σ(y) σ(x + y)−1 σ · · equals

−1 −1 −1 σ2(x2)(σ1(x1)σ2(y2)σ1(x1) σ2(y2) )σ2(x2) . (6.9)

Since σ (x )σ (y )σ (x )−1σ (y )−1 k×, (6.9) equals (6.8). Hence (6.7) holds. 1 1 2 2 1 1 2 2 ∈

6.3.2 Non-degenerate theta groups and vector bundles

Let X be an abelian variety defined over an algebraically closed field k. Let E be a simple vector bundle on X. We have homomorphisms ι : k× G(E) and π : G(E) K(E) defined, re- E → E → spectively, by α (0, α id ) and (x, φ) x. These homomorphisms determine a 7→ E 7→ short exact sequence of groups

ι π 1 k× E G(E) E K(E) 0. → −→ −→ →

Since image ι center G(E), the group G(E) is a central extension of k× by K(E). E ⊆ 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 78

Let E be a simple semi-homogeneous vector bundle of separable type on X. Then K(E) is a ﬁnite group and χ(E)2 = #K(E), [33, 6, 7, and Corollary 7.9, p. § 271]. We prove that G(E) is a non-degenerate theta group.

Lemma 6.10. Let f : X Y be a separable isogeny and let E be a simple semi-homogeneous → ∗ vector bundle of separable type on X. If E ∼= f F , for some vector bundle F on Y , then F is a simple semi-homogeneous vector bundle of separable type on X.

Proof. Mukai’s theory implies that F is semi-homogeneous [33, Proposition 5.4, p.

259]. To prove that F is simple, using Lemma 6.4, we obtain the inequalities

1 = dimk EndOX (E) > dimk EndOY (F ) > 1 and conclude that F is simple.

On the other hand, we have χ(E) = (#ker f)χ(F ) while χ(E)2 = #K(E) and χ(F )2 = #K(F ). We conclude that χ(F ) = 0 and that χ(F ) is not divisible by the 6 characteristic of k.

Proof of Theorem 6.1. Let K be a maximal level subgroup of G(E). (Lemma 6.8 ensures that such a subgroupe exists.) Let K := πE(K). Then K is a maximal subgroup of K(E) on which [ , ] 1. − − G(E) ≡ e Let Y := X/K, let f : X Y denote the quotient map, and let F be a vec- → ∗ tor bundle on Y with the property that f F ∼= E. Then F is a simple semi- homogeneous vector bundle of separable type on X (Lemma 6.10). Using the fact that K is maximal, together with Proposition 6.6, we check that

e G(F )=(k× K)/K · e e 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 79

and conclude that K(F ) is trivial. Since χ(F )2 = #K(F ) we conclude that χ(F )2 = 1. Since χ(E)2 = (#K)2χ(F )2

we conclude that χ(E)2 = (#K)2. Since K was an arbitrary maximal level sub-

2 2 group, we conclude that χ(E) = (#K) fore every maximal level subgroup K of G(E). Let K := x K(E) : [x, y] = 1 for all y K(E) . Then [ , ] induces 0 { ∈ G(E) ∈ } − − G(E) a non-degenerate skew-symmetric bilinear form

K(E)/K K(E)/K k×. 0 × 0 →

2 ′ Hence, #K(E)/K0 = ℓ , for some ℓ, and there exists a maximal subgroup K of K(E)/K , of order ℓ, on which [ , ] 1. (Use Proposition 6.7.) 0 − − G(E) ≡ Let K be the inverse image of K′ in K(E). Then, K is the image of a maximal level subgroup of G(E). We conclude that

χ(E)2 = (#K)2 = (#K )2 ℓ2 0 ·

and hence that

χ(E)2 = (#K )2 ℓ2 = (#K )2 #(K(E)/K )=(#K )#(K(E)) = (#K ) χ(E)2. 0 · 0 · 0 0 0 ·

Hence, #K0 =1 which implies that K0 is trivial.

6.4 Representations of non-degenerate theta groups

Let K be a ﬁnite abelian group and assume that the characteristic of k does not divide the order of K. Let G be a non-degenerate central extension of k× by K. We determine the representation theory of G. 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 80

Before proceeding, we fix some terminology. A G-module (V, ρ) is always a finite dimensional vector space which admits a basis for which there exists Lau- B rent polynomials F k[t, t−1] with the property that the matrix representation of i,j ∈ ρ(α), α k×, with respect to is given by evaluating F at α. ∈ B i,j Theorem 6.11. Let K be finite abelian group and assume that the characteristic of k does not divide the order of K. Let G be a non-degenerate central extension of k× by K of type (d ,...,d ). There exists exactly gcd(n, d )2 gcd(n, d )2 non-isomorphic irreducible 1 p 1 ··· p weight n G-module(s). A weight n representation is irreducible if and only if it has dimension d1···dp . Every weight n G-module decomposes into a direct sum of gcd(n,d1)··· gcd(n,dp) irreducible weight nG-modules. Every G-module decomposes into a direct sum of weight nG-modules.

To determine the representation theory of G, considering Proposition 6.9, it suf-

ﬁces to determine the representation theory of the group Gh−,−i. In what follows we omit the subscript , and denote G simply by G. h− −i h−,−i Let (d ,...,d ) be the type of G and deﬁne D := d1···dp , for each 1 p n gcd(n,d1)··· gcd(n,dp) n Z. ∈

6.4.1 Auxiliary results and proof of Theorem 6.11

Central to our proof of Theorem 6.11 is

Proposition 6.12. If (V, ρ) is a nonzero weight n G-module, then (V, ρ) admits a Dn- dimensional submodule.

Proof. Let (V, ρ) be a weight nG-module. An element x of K acts on a vector v V 1 ∈ by the rule x v := ρ((1, x, 0))(v). We denote the resulting K -module by ResG (V ). · 1 K1 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 81

× G Since K2 = HomZ(K1,k ) the K1-module ResK1 (V ) admits a decomposition into weight spaces. Explicitly, we have

G ResK1 (V )= Vy (6.10) y∈K M2 and if x K , y K , and v V , then x v = x, y v. ∈ 1 ∈ 2 ∈ y · h i Observe now that, if y K , v V , and (α,x,w) G, then ∈ 2 ∈ y ∈

ρ((α,x,w))(v) V . (6.11) ∈ y+nw

Let π : K K denote the group homomorphism deﬁned by y ny. The n 2 → 2 7→ image of πn has order Dn. Using (6.11), we see that every V , y K , appearing in the decomposition y ∈ 2 (6.10), is stable under the (evident) action of ker π . As a consequence if y K n ∈ 2 then the ker πn-module Vy decomposes into weight spaces

Vy = Vy,χ. (6.12) × χ∈HomZM(ker πn,k ) Let σ be a set-theoretic section of the surjective homomorphism K image π 2 → n × induced by π . If z image π , y K , χ HomZ(ker π ,k ), and s V , then n ∈ n ∈ 2 ∈ n y,χ ∈ y,χ deﬁne

sy,χ(z) := ρ((1, 0, σ(z)))(sy,χ).

Observe now that

ρ((α,x,w))(s (z)) = αn x, y + z χ(w + σ(z) σ(nw + z))s (nw + z) (6.13) y,χ h i − y,χ for all (α,x,w) G. ∈ 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 82

× Since (V, ρ) is nonzero, there exists y K and χ HomZ(ker π ,k ) such that ∈ 2 ∈ n V =0. Fix such a pair (y, χ), choose a nonzero vector s V , and deﬁne y,χ 6 y,χ ∈ y,χ

W σ := span s (z) . y,χ k{ y,χ }z∈image πn

σ Using equation (6.13), we see that Wy,χ isa Dn-dimensional G-submodule of V .

Corollary 6.13.

(a) A weight nG-module is irreducible if and only if it has dimension Dn. (b) Every weight nG-module decomposes into weight n irreducible G-modules.

Proof. To prove (a), note that if V is an irreducible weight n G-module then it is nonzero and hence, by Proposition 6.12, admits a submodule of dimension Dn. Since V is irreducible this submodule must equal V whence the dimension of V equals Dn.

Conversely, let V be a weight nG-module of dimension Dn. Let W be a nonzero submodule. By Proposition 6.12, W admits a submodule of dimension Dn. Conse- quently, we have

Dn 6 dim W 6 dim V = Dn.

Hence W has dimension Dn so W = V . We conclude that V is irreducible. To prove (b), let µ k× denote the multiplicative group of d th roots of unity. d1 ⊆ 1 Let G′ denote the subgroup

G′ := (α,x,y) : α µ , x K ,y K { ∈ d1 ∈ 1 ∈ 2} of G. To ﬁnish the proof of Corollary 6.13, we induct on the dimension of V . The base case is dim V =0 in which case the assertion holds. If dim V = N, then combining 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 83

Proposition 6.12 and part (a), which we just proved, we see that V admits an irreducible weight n submodule W . If W = V then we are done. Otherwise, choose a projection p : V W and let p : V W be the projection deﬁned by 0 → →

1 −1 v ′ g p0(g v). 7→ G ′ · · | | gX∈G Then ker p is a G′-submodule of V and V = W ker p. Let s ker p. Then, if ⊕ ∈ (α,x,y) G, we obtain ∈

ρ((α,x,y))(s)= ρ((α, 0, 0))(ρ((1,x,y))(s)) = αn(ρ((1,x,y))(s)).

Since (1,x,y) G′ and since ker p is G′-stable, we conclude ∈

αn(ρ((1,x,y))(s)) ker p. ∈

Hence, ker p is a weight nG-submodule of V . By induction, ker p decomposes into irreducible weight nG-modules.

To construct irreducible weight nG-modules ﬁrst let

× y K , χ HomZ(ker π ,k ), ∈ 2 ∈ n and

W := span e . y,χ k{ y+z,χ}z∈image πn As in the proof of Proposition 6.12, we ﬁx a set-theoretic section σ of the surjective homomorphism K image π induced by π . For every (α,x,w) G deﬁne 2 → n n ∈ an automorphism ρσ,n((α,x,w)) : W W y,χ y,χ → y,χ by e αn x, y + z χ(w + σ(z) σ(nw + z))e y+z,χ 7→ h i − y+z+nw,χ 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 84

and extending linearly.

σ,n Proposition 6.14. The pair (Wy,χ, ρy,χ) is an irreducible weight nG-module.

Proof. It is clear from the deﬁnition that the identity of G acts trivially. Let (α, a, b) and (β,c,d) be elements of G. We then have,

(α, a, b) (β,c,d)=(αβ a, d , a + c, b + d). · h i

By linearity, it sufﬁces to check that

(αβ a, d , a + c, b + d) e =(α, a, b) ((β,c,d) e ). (6.14) h i · y+z,χ · · y+z,χ

σ,n By deﬁnition of ρy,χ the left hand side of equation (6.14) equals

αnβn a, d n a + c,y + z χ(b + d + σ(z) σ(nb + nd + z))e . (6.15) h i h i − y+z+nb+nd,χ

On the other hand,

(β,c,d) e = βn c,y + z χ(d + σ(z) σ(nd + z))e · y+z,χ h i − y+z+nd,χ and

(α, a, b) e = αn a, y + z + nd χ(b + σ(z + nd) σ(nb + z + nd))e . · y+z+nd,χ h i − y+z+nd+nb,χ

Hence, the right hand side of equation (6.14) equals

αnβn c,y + z a, y + z + nd χ(b + d + σ(z) σ(nb + nd + z))e h ih i − y+z+nd+nb,χ which simpliﬁes to equation (6.15). Hence, equation (6.14) holds.

× Since k acts with weight n and since Wy,χ has dimension Dn, we conclude that

σ,n (Wy,χ, ρy,χ) is an irreducible weight nG-module. (Corollary 6.13). 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 85

We now characterize irreducible weight n representations.

Proposition 6.15. A weight n G-module (V, ρ) is irreducible if and only if it is isomor-

σ,n × phic to (W , ρ ) for some y K and some χ HomZ(ker π ,k ). Furthermore y,χ y,χ ∈ 2 ∈ n σ,n σ,n ′ ′ (W , ρ ) is isomorphic to (W ′ ′ , ρ ′ ′ ) if and only if y y image π and χ = χ . y,χ y,χ y ,χ y ,χ − ∈ n

Proof. If (V, ρ) is irreducible, then it equals the subspace

W σ := span s (z) y,χ k{ y,χ }z∈image πn constructed in the proof of Proposition 6.12, for some y K , for some element χ ∈ 2 × of HomZ(ker π ,k ), and for some nonzero vector s V . n y,χ ∈ y,χ Identifying the basis vectors s (z) of W σ with those e { y,χ }z∈image πn y,χ { y+z,χ}z∈image πn σ,n of Wy,χ, and computing the matrix representations of ρ and ρy,χ with respect to

σ,n these bases, we conclude that (V, ρ) is isomorphic to (Wy,χ, ρy,χ).

′ ′ σ,n If y y image π and χ = χ then we conclude that (W ′ ′ , ρ ′ ′ ) is isomorphic − ∈ n y ,χ y ,χ σ,n to (Wy,χ, ρy,χ) by considering their matrix representations with respect to the bases

e ′ and e { y +z,χ}z∈image πn { y+z,χ}z∈image πn reordering one of them if necessary.

σ,n σ,n ′ ′ Conversely, if (Wy,χ, ρy,χ) is isomorphic to (Wy ,χ , ρy′,χ′ ), then they are isomorphic as K1-modules and as ker πn-modules. If they are isomorphic as K1-modules, then every y + z, z ker π equals y′ + z′ for some z′ ker π . In particular, ∈ n ∈ n y y′ ker π . If they are isomorphic as ker π -modules, then χ = χ′. − ∈ n n

Proof of Theorem 6.11. The ﬁrst assertion is a consequence of Proposition 6.15 and a counting argument. The second and third assertions are immediate consequences 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 86

of Corollary 6.13. For the ﬁnal assertion let (V, ρ) be a G-module. Then the k×-

G module Resk× (V ) admits a decomposition

G Resk× (V )= Vn Z Mn∈ into weight spaces. Since the image of k× in G is contained in the center of G each weight space Vn is G-stable and, hence, a weight nG-module.

Remark. Let n be an integer and let r be the remainder obtained by dividing n by

′ d1. Let G be the subgroup

G′ := (α,x,y) : α µ , x K , and y K { ∈ d1 ∈ 1 ∈ 2}

deﬁned in the proof of Corollary 6.13. There is a one-to-one correspondence between irreducible weight n representations of G and irreducible weight r representations of G′.

As a consequence, for a ﬁxed r, 0 6 r 6 d 1, there exists gcd(r, d )2 gcd(r, d )2 1− 1 ··· p non-isomorphic irreducible G′-modules each of which has dimension d1···dp . gcd(r,d1)··· gcd(r,dp) Since,

d1−1 d d 2 G′ = d (d d )2 = gcd(r, d )2 gcd(r, d )2 1 ··· p | | 1 1 ··· p 1 ··· p gcd(r, d ) gcd(r, d ) r=0 1 p X ··· we conclude that G′ has exactly d1−1 gcd(r, d )2 gcd(r, d )2 non-isomorphic ir- r=0 1 ··· p reducible G′-modules and, also,P exactly this number of conjugacy classes. See for example [50, 2.4]. §

Remark. If n =1, then πn is an isomorphism and we may take the set-theoretic sec-

× tion σ to be the identity map. Also, when n =1, every χ HomZ(ker π ,k ) is triv- ∈ n ial. The resulting representation (W , ρσ,1 ) takes the form W := span e y,χ y,χ y,χ k{ z}z∈K2 and an element (α,x,w) acts by (α,x,w) e := α x, z e . · z h i z+w 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 87

6.4.2 Induced representations

We now prove that every irreducible representation is inducedbya 1-dimensional representation of a suitable subgroup. In light of Theorem 6.11, and its proof, it

σ,n × sufﬁces to prove that every (W , ρ ), where y K , χ HomZ(ker π ,k ), and σ y,χ y,χ ∈ 2 ∈ n is a set-theoretic section of πn, is induced by such a representation.

× Fix y K , ﬁx χ HomZ(ker π ,k ), and deﬁne ∈ 2 ∈ n

V := span e . y,χ k{ y,χ}

Let G(ker πn) denote the subgroup of G deﬁned by

G(ker π ) := (α,x,w) : w ker π . n { ∈ n}

We regard Vy,χ as a G(ker πn)-module by deﬁning

(α,x,w) e := αn x, y χ(w)e · y,χ h i y,χ for (α,x,w) G(ker π ). We let ∈ n

ι : V (W , ρσ,n) y,χ → y,χ y,χ denote the G(ker π )-homomorphism deﬁned by e e . n y,χ 7→ y,χ

σ,n G Proposition 6.16. In the above notation, (Wy,χ, ρy,χ) is isomorphic to IndG(ker πn)(Vy,χ).

Proof. We verify the universal property which characterizes induced representations. Let (V, ρ) be a G-module and let Φ : V (V, ρ) be a homomorphism of y,χ → G(ker π )-modules. Deﬁne a map Ψ : W V by n y,χ →

e ρσ,n((1, 0, σ(z))(Φ(e )) y+z,χ 7→ y,χ y,χ 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 88

and extending linearly. Clearly, Ψ is unique and Φ=Ψ ι. It remains to check that ◦ Ψ is a G-homomorphism.

Let z image π and let (α,x,w) G. Using the deﬁnitions of ρσ,n and Ψ, we ∈ n ∈ y,χ obtain that Ψ((α,x,w) e ) equals · y+z,χ

αn x, y + z χ(w + σ(z) σ(nw + z))((1, 0, σ(z + nw)) Φ(e )). (6.16) h i − · y,χ

On the other hand, since Φ is a G(ker π )-homomorphism, Φ(e ) V , where n y,χ ∈ y,n × G V is the (y, n) weight space of the k K -module Res × (V ). y,n × 1 k ×K1×{0} Now, observe

(α,x,w) (1, 0, σ(z)) = (1, 0,w + σ(z)) (α x, σ(z) , x, 0) · · h i

so (α,x,w) Ψ(e ) equals · y+z,χ

(1, 0,w + σ(z)) (αn x, σ(z) n x, y Φ(e )) · h i h i y,χ

which simpliﬁes to

αn x, z + y ((1, 0,w + σ(z)) Φ(e ). (6.17) h i · y,χ

Finally, to see that (6.17) equals (6.16) note that

w + σ(z)= w + σ(z) σ(nw + z)+ σ(z + nw), −

that w + σ(z) σ(nw + z) ker π , and use the fact that Φ is G(ker π )-linear. − ∈ n n

6.4.3 Proof of Theorem 6.2 and Corollary 6.3

Proof of Theorem 6.2. If E is a simple semi-homogeneous vector bundle of separable type then G(E) is a non-degenerate theta group. As a consequence, Theorem 6.2 is

a special case of Theorem 6.11. 6. THETA GROUPS AND VECTOR BUNDLES ON ABELIAN VARIETIES 89

To prove Corollary 6.3 we ﬁrst establish

Proposition 6.17. Let X be an abelian variety. Let E be a non-degenerate simple semi-

homogeneous vector bundle on X. Then E satisﬁes the index condition: there exists a unique integer i(E) with the property that Hi(E)(X, E) =0. In addition, let A be an ample 6 line bundle on X. The roots of the polynomial χ(AN E) are real and i(E) equals the ⊗ number of positive roots counted with multiplicity.

f Proof. By [33, Theorem 5.8, p. 260], E = f (L) for some isogeny Y X, and some ∼ ∗ −→ line bundle L on Y . Let A be an ample line bundle on X and let N be an integer. Using the push-pull formula and the Leray spectral sequence we check that

Hi(X, A⊗N E) = Hi(Y, f ∗(A)⊗N L), for all i. (6.18) ⊗ ∼ ⊗

Using the isomorphisms (6.18), we conclude that

χ(A⊗N E)= χ(f ∗(A)⊗N L). (6.19) ⊗ ⊗

Setting N =0, and using (6.19) and (6.18), we conclude that L is non-degenerate and that Hj(X, E)=0 for all j = i(L). Hence, E satisﬁes the index condition and 6 i(E)=i(L). On the other hand, if N is positive then f ∗(A)⊗N is an ample line bundle on Y .

Using Mumford’s index theorem [41, p. 145] applied to L, together with equation (6.19), we conclude that the roots of the polynomial χ(A⊗N E) are real and that ⊗ i(E) equals the number of positive roots counted with multiplicity.

Proof of Corollary 6.3. We know that G(E) has a unique irreducible weight 1 representation. This representation has dimension equal to # K(E). On the other hand, Hi(E)(X, E) is a weight 1 representation of dimension χp(E) = # K(E). | | p 7

Adelic theta groups and line bundles

Let X be an abelian variety deﬁned over an algebraically closed ﬁeld k. We construct adelic theta groups associated to line bundles on X. We use our construc-

tions to realize the Neron-Severi group of X as a subgroup of H2(V(X); k×). We prove that this realization is functorial in X.

7.1 Formulation and statement of results

Let I denote the set of positive integers which are not divisible by the characteristic of k. Let tor(X) := x X(k) : nx =0 for some n I , { ∈ ∈ } and let V(X) := lim tor(X), where the limit is indexed by I and where the maps ←− are given by n tor(X): tor(X) tor(X) m X | → whenever m divides n.

90 7. ADELIC THETA GROUPS AND LINE BUNDLES 91

Let L be the total space of a line bundle on X. In 7.4, we construct a group § G(L), which we call the adelic theta group of L. It is a central extension of k× by V(X).

Our construction is inspired by Mumford’s approach, [37, 7] and [40, Chapter b § 4], and grew out of efforts to understand these works. Our construction is also different from that of S. Shin [52]. In 7.5, we use our construction to prove § Theorem 7.1. The map G : NS(X) H2(V(X); k×), deﬁned by sending the class of a → line bundle L to that of itsb adelic theta group G(L), is an injective group homomorphism. It satisﬁes the following functorial property: if f : X Y is a homomorphism of abelian b → varieties, then the diagram

G NS(X) / H2(V(X); k×) O b O f ∗ f ∗

G NS(Y ) / H2(V(Y ); k×) b commutes.

7.2 Preliminaries

Conventions about line bundles.

Let X be a projective variety and let L be the total space of a line bundle on

X. If σ is an automorphism of X, then an automorphism of L covering σ is a linear isomorphism τ : L L with the property that the diagram →

τ L / L

σ X / X 7. ADELIC THETA GROUPS AND LINE BUNDLES 92

commutes [41, p. 208]. If f : Y X is a morphism of projective varieties then f ∗L := Y L is deﬁned → ×X by the ﬁbered square

∗ f L := Y X L / L ×

f Y / X. It is the total space of a line bundle on Y . Let X be an abelian variety and let L be the total space of a line bundle on X. If x X then let Aut (L) denote the set of automorphisms of L which cover T . ∈ x x Let K(L) := x X(k) : Aut (L) = ∅ and let G(L) denote the group consist- { ∈ x 6 } ing of pairs (x, φ) where x K(L) and where φ is an automorphism of L covering ∈

Tx. Torsion points.

We identify V(X) with the set n x =(x ) : x tor(X) and x = x if n, m I and m divides n . { i i∈I i ∈ m n m ∈ } We let T(X) := x V(X) : x =0 . This is a subgroup of V(X). { ∈ 1 } Let L be the total space of a line bundle on X. If x V(X), then let ∈ suppL(x) := n I : Aut (n∗ L) = ∅ , { ∈ xn X 6 } and, if m suppL(x), then let ∈ suppL(x) := n I : m divides n . m| { ∈ } A homomorphism of abelian varieties f : Y X induces a homomorphism → V(f):V(Y ) V(X), deﬁned by y =(y ) (f(y )) . → i i∈I 7→ i i∈I We denote V(f)(y) V(X) simply by f(y) in what follows. ∈ 7. ADELIC THETA GROUPS AND LINE BUNDLES 93

Proposition 7.2. Let f : X Y be a homomorphism of abelian varieties. Let L be a line → bundle on Y . The following assertions hold

(a) if y V(Y ) and m is the order of y , then m suppL(y); ∈ 1 ∈ (b) if y V(Y ), m suppL(y), and n suppL(y) , then n suppL(y); ∈ ∈ ∈ m| ∈ (c) if y, z V(Y ), then suppL(z) suppL(y) = ∅; ∈ ∩ 6 ∗ (d) if x V(X), then suppf L(x) suppL(f(x)) = ∅. ∈ ∩ 6

Proof. To prove (a), let y V(Y ) and let m be the order of y . Then y Y 2 and ∈ 1 m ∈ m ∗ ∗ L so Aut (m L) = ∅ because Y 2 K(m L). Hence m supp (y). ym Y 6 m ⊆ Y ∈ n To prove (b), let q = m and note that

n∗ L = q∗ m∗ L = q∗ T ∗ m∗ L = T ∗ (q∗ m∗ L)= T ∗ n∗ L. Y Y Y ∼ Y ym Y yn Y Y yn Y

To prove (c), let y, z V(Y ) and let m be the least common multiple of the order ∈ of y and z . Then, by parts (a) and (b), m suppL(y) suppL(z). 1 1 ∈ ∩ ∗ To prove (d), if x V(X) and if m is the order of x then m suppf L(x) by part ∈ 1 ∈ ∗ (a) applied to X and f L. On the other hand, f(x ) Y 2 so that m is an element m ∈ m of suppL(f(x)) as well.

7.3 Diagramsof k×-torsors

Let L be the total space of a line bundle on X. We construct commutative diagrams of k×-torsors which we use to construct the adelic theta group of L. 7. ADELIC THETA GROUPS AND LINE BUNDLES 94

f ∗ 7.3.1 The pull-back morphisms Aut (L) Autx(f ∗L) f(x) −→ f Let X Y be a homomorphism of abelian varieties. Let L be a line bundle on Y −→ and let x X. We construct pull-back maps ∈ f ∗ Aut (L) Aut (f ∗L) f(x) −→ x

which are morphisms of k×-torsors.

There are two cases to consider. The first is when Autf(x)(L) = ∅. In this case, f ∗ is the empty morphism. The second is when Aut (L) = ∅. In this case, let f(x) 6 φ Aut (L). Then, by the universal property of fibered products, there exists a ∈ f(x) Φ unique morphism L T ∗ L fitting into the commutative diagram −→ f(x)

L E EE φ EE EE Φ E" ∗ % Tf(x)L / L

Tf(x) Y / Y , where the square is a ﬁbered product square and where the unlabeled arrows are the projection morphisms. Since f T = T f, we obtain an isomorphism of ◦ x f(x) ◦ ∗ f L covering Tx via the composition

f ∗Φ f ∗L f ∗T ∗ L = T ∗f ∗L f ∗L, −−→ f(x) x → where the rightmost arrow is the projection map. We denote this isomorphism by f ∗φ. This deﬁnes a map f ∗ Aut (L) Aut (f ∗L), f(x) −→ x which is a morphism of k×-torsors. The following lemma gives an explicit local

description of this pull-back morphism. 7. ADELIC THETA GROUPS AND LINE BUNDLES 95

Lemma 7.3. Let U be an open subset of Y over which L is trivial. Let x X, and let φ be ∈ an element of Autf(x)(L). If

−1 1 1 − φ : Tf(x)(U) A = L|T 1 (U) L|U = U A × f(x) → × is given by (y, t) (y + f(x), αt), with α k×, then 7→ ∈

∗ −1 −1 1 ∗ ∗ −1 1 − − f φ : f (Tf(x)(U)) A = f L|f −1(T 1 (U)) f L|f 1(U) = f (U) A × f(x) → × is given by (a, s) (a + x,αs). 7→ Proof. Let Φ denote the map L T ∗ L determined by φ. Let π : f ∗L X denote → f(x) → the projection, let z f ∗L, and let U be an open subset of Y , over which L is trivial, ∈ and with the property that if V := T −1(f −1(U)) then π(z) V . x ∈ Considering the deﬁnitions, in terms of ﬁbered product squares, of the maps which constitute the composition

∗ 1 ∗ f Φ ∗ ∗ ∗ ∗ −1 1 V A = f L f T L = T f L f L −1 = f (U) A , × |V −−→ f(x) |V x |V → |f (U) × we compute that the image of z = (a, s) V A1 under this composition equals ∈ × (a + x,αs).

Notation. We use the notation

∗ f ∗ Autf(x)(L) Autx(f L) to indicate that Aut (L) = ∅. Note that, in this case, the pull-back map f(x) 6 f ∗ Aut (L) Aut (f ∗L) f(x) −→ x is an isomorphism of k×-torsors. The following proposition shows that the pull-back map just deﬁned behaves well with respect to composition. 7. ADELIC THETA GROUPS AND LINE BUNDLES 96

f g Proposition 7.4. Let X Y Z be a composition of homorphisms of abelian varieties, −→ −→ let L be a line bundle on Z, and let x X. The diagram ∈

∗ (g◦f) ∗ Autg(f(x))(L) / Autx((g f) L) ll6 ◦ lll g∗ lll lll f ∗ ll ∗ Autf(x)(g L) commutes. In otherwords, we have (g f)∗ = f ∗ g∗. ◦ ◦

Proof. Let φ Aut (L) and let z (g f)∗L. It sufﬁces to show that ∈ g(f(x)) ∈ ◦

(g f)∗(φ)(z)= f ∗(g∗(φ))(z). (7.1) ◦

Let π :(g f)∗L X denote the projection map. Let U be an open subset of Z ◦ → z with the property that T −1((g f)−1(U )) is a neighborhood of π(z), and with the xn ◦ z property that L|Uz is trivial. Then the line bundles

∗ ∗ (g f) L −1 , (g f) L −1 −1 , ◦ |(g◦f) (Uz) ◦ |Tx ((g◦f) (Uz))

∗ ∗ g L −1 , g L −1 , L , and L −1 |g (Uz) |T − |(Uz) |T (Uz) f(xm)(g 1(Uz)) g(f(x)) are trivial.

Furthermore, the map

−1 1 1 − φ : Tg(f(x))(U) A = L|T 1 (U) L|U = U A × g(f(x)) → × is given by (c,s) (c + g(f(x)),αs) for some α k×. 7→ ∈ Since

∗ −1 −1 1 z (g f) L −1 −1 = T ((g f) (Uz)) A , ∈ ◦ |Tx ((g◦f) (Uz)) x ◦ × we have that z =(a, t) for some a T −1((g f)−1(U )) and some t A1. Thus, by ∈ x ◦ z ∈ Lemma 7.3, (g f)∗(φ)(z)=(a + x, αt). ◦ 7. ADELIC THETA GROUPS AND LINE BUNDLES 97

On the other hand, let ψ = g∗(φ). Then, by Lemma 7.3, the map

∗ ∗ −1 −1 ψ : g L|T (g−1(U )) g L|g (Uz) f(x) z → is given by (b, r) (b + x,αr). We conclude again, by Lemma 7.3, that 7→ f ∗(ψ)(z)=(a + x, αt).

Hence, equation (7.1) holds.

L,x an,m 7.3.2 The morphisms Autx (m∗ L) Autx (n∗ L) m X −−→ n X Let x V(X), let m suppL(x), let n suppL(x) , and let q := n . We have ∈ ∈ ∈ m| m isomorphisms of k×-torsors

∗ ∗ ∗ Autxn (qX mX L) = Autxn (nX L), (7.2) deﬁned by sending an element φ Aut (q∗ m∗ L) to the element of Aut (n∗ L) ∈ xn X X xn X determined by the composition

φ n∗ L = q∗ m∗ L q∗ m∗ L = n∗ L. X X X −→ X X X Using the isomorphism (7.2), we obtain pull-back morphisms of k×-torsors

q∗ ∗ X ∗ ∗ ∗ Autxm (mX L) Autxn (qX mX L) Autxn (nX L)

L,x which we denote by an,m. These morphisms have the following properties.

Proposition 7.5. Let L be a line bundle on an abelian variety X. Let x V(X) and let ∈ m be an element of suppL(x). The morphisms of k×-torsors

aL,x : Aut (m∗ L) Aut (n∗ L), n,m xm X → xn X deﬁned for all n suppL(x) , have the properties that ∈ m| 7. ADELIC THETA GROUPS AND LINE BUNDLES 98

(a) if x V(X), then aL,x aL,x = aL,x, for all n, m, p suppL(x) with the property ∈ n,m ◦ m,p n,p ∈ that p m n; | | L,x (b) if x V(X), then a = idAut (n∗ L); ∈ n,n xn X (c) if x and y are elements of V(X), then

aL,x+y(φ ψ)= aL,x (φ) aL,y (ψ), n,m ◦ n,m ◦ n,m

for all φ Aut (m∗ L) and all ψ Aut (m∗ L) whenever ∈ xm X ∈ ym X

m suppL(x) suppL(y). ∈ ∩

Proof. Assertions (a) and (b) are consequences of Proposition 7.4 and Lemma 7.3

respectively.

n To prove (c), let q := m . It sufﬁces to check that the diagram

◦ Aut (n∗ L) Aut (n∗ L) / Aut (n∗ L) xn X × yn X xn+yn X

◦ Aut (q∗ m∗ L) Aut (q∗ m∗ L) / Aut (q∗ m∗ L) xn X X × yn X X xn+yn X X ∗ ∗ ∗ qX ×qX qX ◦ Aut (m∗ L) Aut (m∗ L) / Aut (m∗ L) xm X × ym X xm+ym X commutes; this can be achieved by direct calculations.

Explicitly, let φ Aut (m∗ L), let ψ Aut (m∗ L), and let π : n∗ L X ∈ xm Y ∈ ym Y n X → denote the projection map. Let z n∗ L, and let U be an open subset such that ∈ X z m∗ L is trivial and with the property that π (z) T −1 (q−1(U )). X |Uz n ∈ xn+yn X z Then, the map

∗ ∗ ψ : m L −1 −1 m L −1 X |Tym (Txm (Uz)) → X |Txm (Uz) is of the form (a, t) (a + y , αt), for some α k×, the map 7→ m ∈

∗ ∗ φ : m L −1 m L|U X |Txm (Uz) → X z 7. ADELIC THETA GROUPS AND LINE BUNDLES 99

is of the form (b, s) (b + x ,βs), for some β k×, and the map 7→ m ∈

∗ ∗ φ ψ : m L −1 −1 m L −1 ◦ X |Tym (Txm (Uz)) → X |Txm (Uz)

is of the form (a, t) (a + x + y ,αβt). 7→ m m By Lemma 7.3, if

∗ −1 −1 1 −1 −1 z =(c,r) nX L|T (q (U )) = Txn+yn (qX (Uz)) A ∈ xn+yn X z × then aL,x+y(φ ψ)(z)=(c + x + y ,αβr). n,m ◦ n n On the other hand, again by Lemma 7.3,

aL,x (φ) aL,y (ψ)(z)= aL,x (φ)(c + y , αt)=(c + x + y ,βαt). n,m ◦ n,m n,m n n n

We conclude that aL,x+y(φ ψ)= aL,x (φ) aL,y (ψ) n,m ◦ n,m ◦ n,m so (c) holds.

f,L bx,m 7.3.3 The morphisms Autx (m∗ f ∗L) Aut (m∗ L) m X −−→ f(xm) Y Let f : X Y be a homomorphism of abelian varieties. Let L be a line bundle → on Y , let x V(X), and let m suppL(f(x)). Since m suppL(f(x)) we have that ∈ ∈ ∈ Aut (m∗ L) = ∅. f(xm) Y 6 We now construct isomorphisms

f,L bx ∗ ,m ∗ ∗ Autf(xm)(mY L) Autxm (mX f L) (7.3) of k×-torsors. 7. ADELIC THETA GROUPS AND LINE BUNDLES 100

Since m f = f m , we have isomorphisms Y ◦ ◦ X

∗ ∗ ∗ ∗ Autxm (f mY L) = Autxm (mX f L).

Inverting the pullback morphism

∗ ∗ f ∗ ∗ Autf(xm)(mY L) Autxm (f mY L) yields the isomorphism (7.3).

f,L The morphisms bx,m have the following properties.

Proposition 7.6. Let f : X Y be a homomorphism of abelian varieties. Let L be a line → bundle on Y . The morphisms bf,L : Aut (m∗ f ∗L) Aut (m∗ L), deﬁned for all x V(X) and m suppL(f(x)), x,m xm X → f(xm) Y ∈ ∈ have the properties that

∗ (a) if x V(X) and n suppf L(x) = suppL(y) then the diagram ∈ ∈ m| m|

f∗L,x ∗ ∗ an,m ∗ ∗ Autxm (mX (f L)) Autxn (nX (f L))

f,L f,L bx,m bx,n

L,f(x) ∗ an,m ∗ Autf(xm)(mY L) Autf(xn)(nY L)

commutes;

∗ ∗ (b) if x, z V(X), m suppf L(x) suppf L(z), φ Aut (m∗ (f ∗L)), and ψ is an ∈ ∈ ∩ ∈ xm X ∗ ∗ element of Autzm (mX (f L)), then

bf,L (φ ψ)= bf,L (φ) bf,L (ψ). x+z,m ◦ x,m ◦ z,m 7. ADELIC THETA GROUPS AND LINE BUNDLES 101

n Proof. We first prove (a). Let q := m . Considering the definitions, we conclude that it suffices to show that the diagram

q∗ ∗ ∗ X ∗ ∗ ∗ ∗ ∗ Autxm (mX f L) Autxn (qX mX f L) Autxn (nX f L)

∗ ∗ ∗ ∗ Autxm (f mY L) Autxn (f nY L)

f ∗ f ∗ q∗ ∗ Y ∗ ∗ ∗ Autf(xm)(mY L) Autf(xm)(qY mY L) Autf(xn)(nY L)

commutes. This can be veriﬁed locally in a manner similar to Lemma 7.3 and Proposition 7.5. To prove (b), it sufﬁces to check that the diagram

◦ Aut (m∗ f ∗L) Aut (m∗ f ∗L) / Aut (m∗ f ∗L) xm X × zm X xm+zm X

◦ Aut (f ∗m∗ L) Aut (f ∗m∗ L) / Aut (f ∗m∗ L) xm Y × zm Y xm+zm Y f ∗×f ∗ f ∗

◦ Aut (m∗ L) Aut (m∗ L) / Aut (m∗ L) f(xm) Y × f(zm) Y f(xm)+f(zm) Y commutes. This can be checked locally, using Lemma 7.3, in a manner similar to statement (c) of Proposition 7.5.

7.4 Adelic theta groups

Let X be an abelian variety and let L be a line bundle on X. We construct G(L), the adelic theta group of L, and discuss some ﬁrst properties. b 7. ADELIC THETA GROUPS AND LINE BUNDLES 102

7.4.1 The group G(L)

Let G(L) denote the set of pairs (x, α L ), where b { n}n∈supp (x) b x V(X), α Aut (n∗ L), and ax,L (α )= α ∈ n ∈ xn X n,m m n whenever m suppL(x) and n suppL(x) . ∈ ∈ m| Observe that G(L) is nonempty.

Lemma 7.7. Let xb V(X), let p suppL(x), and let α be an element of Aut (p∗ L). ∈ ∈ p xp X For l suppL(x), deﬁne ∈

L,x L al,p (αp) if l supp (x)p| αl :=  ∈ (7.4)  L,x −1 L,x L a (a (αp)) if l supp (x)p| . lp,l lp,p 6∈  Then (x, αn n∈suppL(x)) is an element of G(L). { } Proof. Let m and n be elements of suppLb(x) and assume that m n. We need to show | L,x+y that an,m (αm) = αn. To reduce notation we omit the superscripts. Considering (7.4), and using the fact that m divides n, we see that there are three cases to consider: p m and p n; p ∤ m and p ∤ n; p ∤ m and p n. We make extensive use of | | | statement (a) of Proposition 7.5.

In the ﬁrst case, we have αn := an,p(αp) and αm := am,p(αp). We then have, since m n, |

an,p(αp)= an,m(am,p(αp))

so that αn = an,m(αm).

−1 −1 In the second case, we have αm := amp,m(amp,p(αp)) and αn := anp,n(anp,p(αp)). Hence,

anp,n(αn)= anp,p(αp)= anp,mp(amp,p(αp)) = anp,mp(amp,m(αm)) = anp,m(αm) 7. ADELIC THETA GROUPS AND LINE BUNDLES 103

−1 from which we deduce αn = anp,n(anp,nan,m(αm)) = an,m(αm).

−1 In the third case, we have αn := an,p(αp) and αm := amp,m(amp,p(αp)). We then observe,

anp,m(αm)= anp,mp(amp,m(αm)) = anp,mp(amp,p(αp)).

Hence,

anp,n(an,m(αm)) = anp,m(αm)= anp,p(αp).

On the other hand, we have

anp,n(αn)= anp,n(an,p(αp)) = anp,p(αp).

We conclude that an,m(αm)= αn.

We now prove that G(L) is a group.

Proposition 7.8. If (x, bα ) and (y, β ) G(L), then let { n}n∈supp(x) { m}m∈supp(y) ∈ b (x, α ) (y, β ):=(x + y, γ ) (7.5) { n}n∈supp(x) · { m}m∈supp(y) { l}l∈supp(x+y) where γ is defined by choosing some element p of suppL(x) suppL(y), which { l}l∈supp(x+y) ∩ is nonempty and contained in suppL(x + y), defining γ := α β , which is an element p p ◦ p of Aut (p∗ L), and defining, for all l suppL(x + y), xp+yp X ∈

L,x+y L al,p (γp) if l supp (x + y)p| γl :=  ∈  L,x+y −1 L,x+y L a (a (γp)) if l supp (x + y)p|. lp,l lp,p 6∈  The right hand side of (7.5) is a well deﬁned element of G(L), the pair (G(L), ) is a · central extension of k× by V(X), and the group (G(L), ) contains an isomorphic copy of · b b T(X). b 7. ADELIC THETA GROUPS AND LINE BUNDLES 104

Proof. Throughout the proof, we make extensive use of Proposition 7.5. Note that Lemma 7.7 implies that the right hand side of (7.5) is an element of G(L). We now show that the right hand side of (7.5) is well-defined. b Let (x, α ), (y, β ) G(L), { n}n∈supp(x) { m}m∈supp(y) ∈ let p, q suppL(x) suppL(y), and define γ := α β andb ζ := α β . ∈ ∩ p p ◦ p q q ◦ q If l suppL(x + y), then define ∈

L,x+y al,p (γp) if p l γl :=  |  L,x+y −1 L,x+y alp,l (alp,p (γp)) if p ∤ l.

  L,x+y al,q (ζq) if q l ζl :=  |  L,x+y −1 L,x+y alq,l (alq,q (ζq)) if q ∤ l. We have to show that γ = ζ. l l There are two cases and several sub-cases to consider. The ﬁrst case is p q. In | this case, we have

ζ := α β = aL,x(α ) aL,y(β )= aL,x+y(α β ). q q ◦ q q,p p ◦ q,p p q,p p ◦ p We conclude that

L,x+y ζq = aq,p (γp).

If q l, then ζ = aL,x+y(ζ ). Since p q and q l we conclude that p l. Hence, we | l l,q q | | | have

L,x+y L,x+y L,x+y L,x+y ζl = al,q (ζq)= al,q (aq,p (γp)) = al,p (γp).

If q ∤ l, then

L,x+y −1 L,x+y L,x+y −1 L,x+y L,x+y L,x+y −1 L,x+y ζl := alq,l (alq,q (ζq)) = alq,l (alq,q (aq,p (γp))) = alq,l (alq,p (γp)). 7. ADELIC THETA GROUPS AND LINE BUNDLES 105

Now if p l, then γ = aL,x+y(γ ). Since aL,x+y is injective and | l l,p p ql,l

L,x+y L,x+y L,x+y L,x+y aql,p (γp)= aql,l (al,p (γp)) = aql,l (γl),

we conclude that ζl = γl.

L,x+y −1 L,x+y If p ∤ l, then γl := alp,l (alp,p (γp)). Hence, we have

L,x+y L,x+y L,x+y L,x+y L,x+y L,x+y aql,l (γl)= aql,pl (apl,l (γl)) = aql,pl (apl,p (γp)) = aql,p (γp).

L,x+y L,x+y L,x+y On the other hand, aql,l (ζl) = alq,p (γp). Since aql,l is injective, we conclude that γl = ζl. The second case is p ∤ q. We then have

ζ := α β = aL,x −1(aL,x (α )) aL,y −1(aL,y (β )), q q ◦ q qp,q qp,p p ◦ qp,q qp,p p and deduce that

L,x+y −1 L,x+y ζq = aqp,q (aqp,p (γp)).

If q l, then |

L,x+y L,x+y L,x+y −1 L,x+y L,x+y ζl = al,q (ζq)= al,q (aqp,q (aqp,p (γp))) = al,q (γq).

If p l, then γ = aL,x+y(γ ). Hence, we have | l l,p p

L,x+y L,x+y L,x+y L,x+y L,x+y L,x+y alpq,l (ζl)= alpq,l (al,q (γq)) = alpq,l (al,p (γp)) = alpq,l (γl),

from which we conclude that ζl = γl.

L,x+y −1 L,x+y If p ∤ l, then γp = apl,p (apl,l (γp)). We then compute

L,x+y L,x+y apql,p (γp)= apql,p (ζp),

and conclude that γp = ζp. 7. ADELIC THETA GROUPS AND LINE BUNDLES 106

If q ∤ l, then ζ := aL,x+y −1(aL,x+y(ζ )). If p l, then γ = aL,x+y(γ ). We l lq,l lq,q q | l l,p p L,x+y L,x+y L,x+y −1 L,x+y then compute alpq,l (ζl) = alpq,l (γl). If p ∤ l, then γl = alp,p (alp,p (γp)). We L,x+y L,x+y compute again that alpq,l (γl)= alpq,l (ζl). We now show that the pair (G(L), ) is a group. The identity element is ·

b L,0 (0, a (id ) L ). { n,1 L }n∈supp (0)

−1 If (x, α L ) G(L) then its inverse is the element ( x, α L ) of { p}p∈supp (x) ∈ − { p }p∈supp (−x) −1 G(L) where αp is theb inverse of αp. b It remains to show that multiplication is associative. Let

(x, α L ), (y, β L ), and (z, γ L ) G(L). { n}n∈supp (x) { m}m∈supp (y) { l}l∈supp (z) ∈ b Since the operation is well-deﬁned it sufﬁces to choose a ·

q suppL(x) suppL(y) suppL(z) ∈ ∩ ∩ and note that (α β ) γ = α (β γ ). q ◦ q ◦ q q ◦ q ◦ q We now construct a short exact sequence

1 k× G(L) V(X) 0 → → → → b with the property that the image of k× in G(L) is contained in the center of G(L).

× L,0 The inclusion k G(L) is deﬁned by α (0, a (id ) 0 ), while the → b 7→ { n,1 L }n∈supp( ) b surjection G(L) V(X) is deﬁned by (x, α L ) x. The map → b { p}p∈supp (x) 7→

b L,x x (x, a (id ) L ) 7→ { n,1 L }n∈supp (x) deﬁnes a (group theoretic) section of this surjection over T(X). 7. ADELIC THETA GROUPS AND LINE BUNDLES 107

7.4.2 The skew-symmetric bilinear form [ , ] − − G(L)

Let (x, α L ) and (y, β L ) be elements of G(b L). If p is an element { n}n∈supp (x) { n}n∈supp (y) of suppL(x) suppL(y), then α β α−1 β−1 corresponds to a unique γ k× ∩ p ◦ p ◦ p ◦ p b ∈ which is independent of our choice of p. Also, if [ , ] denotes the commutator of G(L), then − − G(L) b [(x, αn L ), (y, βn L )] b=(0, γ idn∗ L L ) { }n∈supp (x) { }n∈supp (y) G(L) { X }n∈supp (0) b and we obtain a skew-symmetric bilinear form

[ , ] : V(X) V(X) k×, deﬁned by [x, y] = γ. − − G(L) × → G(L) b b Observe also that, if x, y V(X), if p suppL(x) suppL(y), and if ∈ ∈ ∩

∗ ∗ × [ , ]G(p∗ L) : K(p L) K(p L) k − − X X × X → denotes the skew-symmetric bilinear form determined by the commutator of the

∗ x y ∗ theta group G(pX L), then we have [ , ]G(L) = [xp,yp]G(pX L). b

7.4.3 The group homomorphism G(f)

Let f : Y X be a homomorphism of abelian varieties. We construct a group → b homomorphism G(f): G(f ∗L) G(L) which ﬁts into the commutative diagram →

b 1 b / k× b/ G(f ∗L) / V(Y ) / 0 (7.6)

G(f) V(f) b b 1 / k× / G(L) / V(X) / 0.

∗ In other words, the extension determinedb by G(f L) is equivalent to the pull-back, V(f) with respect to the group homomorphism V(Y ) V(X), of the extension deter- b −−→ mined by G(L).

b 7. ADELIC THETA GROUPS AND LINE BUNDLES 108

∗ Let (x, α f∗L ) G(f L). Let m be the order of x and let y := f(x). { n}n∈supp (x) ∈ 1 ∗ Then m suppf L(x) suppL(y). ∈ ∩ b Let β := bf,L (α ) and, for p suppL(y), deﬁne m x,m m ∈

L,y L ap,m(βm) if p supp (y)m| βp :=  ∈  L,y −1 L,y L a a (βm) if p supp (y)m|. pm,p pm,m 6∈  Let 

G(f)((x, α f∗L )) := (y, β L ). { n}n∈supp (x) { n}supp (y) Using Propositionb 7.6, we check that the above definition defines a group homomorphism G(f) : G(f ∗L) G(L) fitting into the commutative diagram (7.6). → b b b 2 (×The group homomorphism NS(X) ֒ H (V(X); k 7.5 → Proof of Theorem 7.1. If L is a line bundle on X, then we let [G(L)] denote the class

2 × of its adelic theta group in H (V(X); k ) the group of normalizedb two cocycles modulo crossed homomorphisms. Step 1. Let L and M be line bundles on X. The relation

[G(L M)]=[G(L)]+[G(M)] ⊗ holds in H2(V(X); k×). b b b To prove Step 1, we deﬁne normalized set-theoretic sections

σ : V(X) G(L), σ : V(X) G(M), and σ : V(X) G(L M) L → M → L⊗M → ⊗ and check that the relationb b b

[ , ] = [ , ] + [ , ] − − σL⊗M − − σL − − σM 7. ADELIC THETA GROUPS AND LINE BUNDLES 109

holds amongst the corresponding factor sets. To establish this, we must show that if x and y are elements of V(X), if

−1 × σL(x) σL(y) σL(x + y) =(0, α idn∗ L L ), α k , · · { X }n∈supp (0) ∈ if

−1 × σM (x) σM (y) σM (x + y) =(0, β idn∗ M M 0 ), β k , · · { X }n∈supp ( ) ∈ and if

−1 × σL⊗M (x) σL⊗M (y) σL⊗M (x + y) =(0, γ idn∗ (L⊗M) L⊗M 0 ), γ k · · { X }n∈supp ( ) ∈ then γ = αβ. Observe that this holds if γ idn∗ (L⊗M) = α idn∗ L β idn∗ M for some n. X X ⊗ X

We now deﬁne sections σL, σM and σL⊗M with the desired properties. First of

all, for any element x of V(X), let mx be the order of x1. Now, let x V(X). If x =0, then deﬁne αx := id and βx := id . Otherwise, ∈ mx L mx M choose αx Aut (m∗ L), choose βx Aut (m∗ M) and set mx ∈ xmx x X mx ∈ xmx x X

γx := αx βx Aut (m∗ (L M)). mx mx ⊗ mx ∈ xmx x X ⊗

x x x Then, by Lemma 7.7, αmx , βmx and γmx determine (unique) elements of G(L), G(M), and G(L M) and hence allow us to deﬁne normalized sections σ , σ ⊗ Lb M andb σL⊗M . b

Now, let x and y be elements of V(X) and let p := lcm(mx, my). Let

φ Aut (p∗ L), ψ Aut (p∗ M) and η Aut (p∗ (L M)) p ∈ 0 X p ∈ 0 X p ∈ 0 X ⊗

be such that

(0,φ )=(x , αx) (y , αy) (x + y , αx+y)−1 G(p∗ L), p p p · p p · p p p ∈ X 7. ADELIC THETA GROUPS AND LINE BUNDLES 110

(0, ψ )=(x , βx) (y , βy) (x + y , βx+y)−1 G(p∗ M), and p p p · p p · p p p ∈ X (0, η )=(x ,γx) (y ,γy) (x + y ,γx+y)−1 G(p∗ (L M)). p p p · p p · p p p ∈ X ⊗

These are the “pth components” of [x, y]σL , [x, y]σM and [x, y]σL⊗M . Since

γx = αx βx,γy = αy βy and γx+y = αx+y βx+y, p p ⊗ p p p ⊗ p p p ⊗ p computing the above multiplications we conclude that η = φ ψ . p p ⊗ p Step 2: If L is a line bundle on X, then G(L) is abelian if and only if L Pic0(X). ∈ 0 Assume that G(L) is abelian. To proveb that L is an element of Pic (L) we relate [ , ] to the Weil-pairing. Let n be a positive integer not divisible by the − − G(L) b characteristicb of k. By [41, 20, p. 170], for instance, there exists a non-degenerate § pairing e : X X µ , n n × n → n

where µn is the group of n-th roots of unityb of k. In addition, if x Xn, y ∈ ∈ n−1(K(L)), and z X is such that nz = y, then X ∈

en(x, φL(y))=[x, z]G(n∗L), by [41, p. 212]. We now prove that if [x, y] =1, for all x, y V(X), then φ (y)= for all G(L) ∈ L OX y X. b ∈ Since tor(X) is Zariski dense in X it sufﬁces to show that φ (y) = for all L OX y tor(X). Let n be the order of y. Then y K(n∗ L), and so y n−1(K(L)). ∈ ∈ X ∈ X Choose z V(X) such that z = y. Then nz = y and also n2z = ny = 0 so ∈ 1 n n z K(n∗ L). Hence, n supp(z). n ∈ X ∈ Let x X and choose x T(X) with x = x. We then have ∈ n ∈ n

∗ en(x, φL(y))=[x, zn]G(n L) = [x, z]G(L) =1, (7.7) b 7. ADELIC THETA GROUPS AND LINE BUNDLES 111

where the second rightmost equality follows because n supp(x) supp(z), and ∈ ∩ where the rightmost equality follows because G(L) is abelian.

Since x is an arbitrary element of X the relation (7.7) holds for all x X . n b ∈ n Since e is non-degenerate this means that φ (y) = which is what we wanted n L OX to show. The above implies that if G(L) is abelian then L Pic0(X). Indeed if G(L) is ∈ abelian then [x, y] =1 for all x, y V(X). This implies that φ (y)= for all G(L) b ∈ L OXb y X. Hence L Pic0(X). ∈ ∈ Conversely, if L Pic0(X), then G(L) is abelian which implies that G(L) is ∈ abelian. b .Step 3. The homomorphism G : NS(X) ֒ H2(V(X); k×) is functorial in X → Let f : X Y be a homomorphism of abelian varieties. Using the deﬁnition of → b G(f) (see 7.4.3) we check that the diagram (which has exact rows) §

b G 1 / Pic0(X) / Pic(X) / H2(V(X); k×) O O b O f ∗ f ∗ f ∗

G 1 / Pic0(Y ) / Pic(Y ) / H2(V(Y ); k×) b commutes. 8

Topics for future investigation

The results of this thesis prompt several new related problems which are worthy of further study. We indicate some of these here.

The central theme of our problems is that we should be able to reﬁne many of the results described in Chapters 4, 5, 6, and 7 especially complex analytically. Throughout, we let Λ Cg denote a lattice and X := Cg/Λ. ⊆

8.1 More general instances of the pair index condition

Theorem 5.1 gives a description of the pair index condition as it applies to line bundles on a particular class of abelian varieties with real multiplication. On the other hand Theorem 4.1 address the asymptotic nature of the pair index condition as it applies to vector bundles on an arbitrary abelian variety. There are several related directions in which these theorems should be improved. To describe such a direction, let us ﬁrst point out that there is a functorial isomorphism between NS(X), the Neron-Severi group of X, and the group consisting

112 8. TOPICS FOR FUTURE INVESTIGATION 113

of Hermitian forms H : Cg Cg C whose imaginary parts are integral on Λ × → [6, Appell-Humbert Theorem 2.2.3 p. 32]. As a consequence, the rational Neron-

Severi group NS(X)Q := NS(X) Q is functorially isomorphic to the group of ⊗ Hermitian forms H : Cg Cg C whose imaginary parts are rational on Λ. × → Let E be a vector bundle on X and let r(E) denote its rank. By the Appel-

Humbert theorem det(E) ∼= L(H, χ) for some uniquely determined AH-data (H, χ). 1 We deﬁne slope(E) := H NS(X)Q. r(E) ∈ In this setting, we pose

Problem 8.1. Let X be a simple complex abelian variety. For what p, q N and ∈ what d, e NS(X)Q does X admit simple vector bundles E and F with the prop- ∈ erties that:

(a) slope(E)= d, slope(F )= e; (b) the pair (E, F ) satisﬁes the PIC; (c) i(E)= p and i(F )= q.

Problem 8.1 should be tractable in the case that X has multiplication by a number ﬁeld. Furthermore, its solution would give a generalization of Theorem 5.1.

One reason that Problem 8.1 should admit a reasonably general solution is related to our work in Chapter 6 and the work of Matsushima [31], which should be seen as the complex analytic analogue to Mukai’s algebraic constructions [33]. More precisely, Matsushima used factors of automorphy to construct simple

vector bundles with prescribed rational class in NS(X)Q [31, Corollary 8.3, p. 184]. He also computed the ﬁrst Chern class of these bundles explicitly [31, Theorem 9.3, p. 195]. A few points, related to Problem 8.1 and Matsushima’s construction, which 8. TOPICS FOR FUTURE INVESTIGATION 114

should be checked are: that this prescribed rational class is the slope (in the sense that we have de- • ﬁned) of the bundles he constructs; these bundles are semi-homogeneous in the sense of Mukai [33]. • Combining this circle of ideas with those of Theorem 5.1 should yield an answer to Problem 8.1 in the case that X has multiplication by a number ﬁeld.

8.2 Embeddings of abelian varieties, pull-backs, and

harmonic forms

Valdemar Tsanov and I have recently formulated problems, related to the results of Chapter 4, which warrant future investigation.

As one example, we want to understand pull-back maps arising from embeddings of complex abelian varieties. We are especially interested in the extent to which these maps can be understood at the level of harmonic forms. We are also interested in understanding moduli problems related to embeddings of complex tori; these moduli problems may be related to the pullback maps we are considering. The problems that we have formulated are motivated by Valdemar’s previous work on semi-simple complex Lie groups [54], my own work on abelian varieties, and the joint work of our (respective) PhD advisers I. Dimitrov and M. Roth [10]. We now make precise formulations of some of the problems that we are considering. Let X and Y be abelian varieties, and let L be a non-degenerate line bundle on Y . Let ι : X ֒ Y be an embedding of abelian varieties. Assume that ι∗L is → 8. TOPICS FOR FUTURE INVESTIGATION 115

non-degenerate and that i(ι∗L)=i(L). A ﬁrst generalization, very much in the spirit of the problems considered in

[54], is to study the pull-back

∗ Hi(L)(Y, L) Hi(ι L)(X, ι∗L). (8.1) →

As in Problem 1.4, a ﬁrst question might be to ask if the map (8.1) is nonzero. As a ﬁrst step to studying this question note that Theorem 4.2 determines the asymptotic nature of these maps. Using Theorem 4.2, we can deduce that these maps are asymptotically surjective and hence asymptotically nonzero. Indeed if denotes the ideal sheaf of the IX image of X in Y then, since ι is an embedding, the map (8.1) is surjective if

Hi(L)+1(Y, L)=0. IX ⊗

On the other hand, Theorem 4.2 implies that Hi(L)+1(Y, Ln)=0 for all n 0 IX ⊗ ≫ whence the restriction map (8.1) is asymptotically surjective. Having said this, it is not entirely obvious the extent to which the pull-back map (8.1) can be studied at the level of harmonic forms. More precisely, let q denote the index of L, let 0,q(Y, L) be the space of (0, q)-harmonic forms of L, and let H 0,q(X, ι∗L) be the space of (0, q)-harmonic forms of ι∗L. H We pose

f ∗ Problem 8.2. In the above situation, does the pull back map 0,q(Y, L) 0,q(X, ι∗L) A −→A induce a map 0,q(Y, L) 0,q(X, ι∗L)? H → H Bibliography

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