Milne - Introduction to Shimura Varieties

8/11/2019 Milne - Introduction to Shimura Varieties

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Introduction to Shimura Varieties

J.S. Milne

October 23, 2004

Abstract

This is an introduction to the theory of Shimura varieties, or, in other words, to thearithmetic theory of automorphic functions and holomorphic automorphic forms.

Contents

Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 Hermitian symmetric domains 8

Brief review of real manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Brief review of hermitian forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Hermitian symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

The three types of hermitian symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . 12

Example: Bounded symmetric domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Automorphisms of a hermitian symmetric domain . . . . . . . . . . . . . . . . . . . . . . . . . . 14

The homomorphismupW U1! Hol(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Cartan involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Representations ofU1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Classification of hermitian symmetric domains in terms of real groups . . . . . . . . . . . . . . . 19

Classification of hermitian symmetric domains in terms of dynkin diagrams . . . . . . . . . . . . 21

2 Hodge structures and their classifying spaces 23

Reductive groups and tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Flag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

The projective space P(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Grassmann varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Flag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

The hodge filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Copyright c 2004 J.S. Milne

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2 CONTENTS

Hodge structures as representations ofS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26The Weil operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Hodge structures of weight0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Tensor products of hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Morphisms of hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Hodge tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Variations of hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Locally symmetric varieties 32

Quotients of hermitian symmetric domains by discrete groups . . . . . . . . . . . . . . . . . . . . 32

Subgroups of finite covolume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Arithmetic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Brief review of algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Algebraic varieties versus complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

The functor from nonsingular algebraic varieties to complex manifolds . . . . . . . . . . . . 36Necessary conditions for a complex manifold to be algebraic . . . . . . . . . . . . . . . . . 37

Projective manifolds and varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

The theorem of Baily and Borel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

The theorem of Borel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Finiteness of the group of automorphisms ofD() . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Connected Shimura varieties 42

Congruence subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Connected Shimura data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Definition of a connected Shimura variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

The strong approximation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

An adelic description ofD() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Alternative definition of connected Shimura data . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Shimura varieties 51

Notations for reductive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

The real points of algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Shimura data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Morphisms of Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

The structure of a Shimura variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Zero-dimensional Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Additional axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Arithmetic subgroups of tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Passage to the limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 The Siegel modular variety 65

Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Symplectic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

The Shimura datum attached to a symplectic space . . . . . . . . . . . . . . . . . . . . . . . . . 66

The Siegel modular variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Complex abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

A modular description of the points of the Siegel variety . . . . . . . . . . . . . . . . . . . . . . 72

7 Shimura varieties of hodge type 73


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8 PEL Shimura varieties 76

Algebras with involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Symplectic modules and the associated algebraic groups . . . . . . . . . . . . . . . . . . . . . . 77

Algebras with positive involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

PEL data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80PEL Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

PEL modular varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9 General Shimura varieties 84

Abelian motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Shimura varieties of abelian type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Classification of Shimura varieties of abelian type . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Shimura varieties not of abelian type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Example: simple Shimura varieties of typeA1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

10 Complex multiplication: the Shimura-Taniyama formula 88

Where we are headed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Review of abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

CM fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Abelian varieties of CM-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Abelian varieties over a finite field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

The Shimura-Taniyama formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

The OE -structure of the tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Sketch of the proof the Shimura-Taniyama formula . . . . . . . . . . . . . . . . . . . . . . 96

11 Complex multiplication: the main theorem 97

Review of class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Convention for the (Artin) reciprocity map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

The reflex field and norm of a CM-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Statement of the main theorem of complex multiplication . . . . . . . . . . . . . . . . . . . . . . 99

12 Definition of canonical models 101

Models of varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

The reflex field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Special points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

The homomorphismrx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Definition of a canonical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Examples: Shimura varieties defined by tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

CM-tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

13 Uniqueness of canonical models 107

Extension of the base field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Uniqueness of canonical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

The galois action on the connected components . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

14 Existence of canonical models 110

Descent of the base field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

The regularity condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

The continuity condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A sufficient condition for descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Review of local systems and families of abelian varieties . . . . . . . . . . . . . . . . . . . . . . 111

The Siegel modular variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

The reflex field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

The special points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112


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A criterion to be canonical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Outline of the proof of the existence of a canonical model . . . . . . . . . . . . . . . . . . . 114

Condition (a) of (14.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Condition (b) of (14.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Condition (c) of (14.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Simple PEL Shimura varieties of type A or C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Shimura varieties of hodge type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Shimura varieties of abelian type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

General Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Final remark: rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

15 Abelian varieties over finite fields 118

Semisimple categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Division algebras; the Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Abelian varieties over Fq ,qD pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Abelian varieties over F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Tori and their representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Affine extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

The affine extensionP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

The local formPl ofP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

The Q`-space attached to a fake motive . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126The isocrystal of a fake motive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Abelian varieties of CM-type and fake abelian varieties . . . . . . . . . . . . . . . . . . . . 127

16 The good reduction of Shimura varieties 129

The points of the Shimura variety with coordinates in the algebraic closure of the rational numbers 129

The points of the Shimura variety with coordinates in the reflex field . . . . . . . . . . . . . . . . 129

Hyperspecial subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

The good reduction of Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Definition of the Langlands-Rapoport set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Definition of the setS () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Definition of the groupI(). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Definition ofXp(). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Definition ofXp(). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Definition of the Frobenius element . . . . . . . . . . . . . . . . . . . . . . . . . 133

The setS (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

The admissibility condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

The condition at 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134The condition at ` 6D p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134The condition atp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

The global condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134The Langlands-Rapoport set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

The conjecture of Langlands and Rapoport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

17 A formula for the number of points 136

Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

The triple attached to an admissible pair (, ") . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

The formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

18 Endnotes 139

Proof of Theorem 5.4 (footnote 39) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Proof of the claim in 5.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

References 142


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Index of definitions 147


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Introduction

The arithmetic properties of elliptic modular functions and forms were extensively studied

in the 1800s, culminating in the beautiful Kronecker Jugendtraum. Hilbert emphasized theimportance of extending this theory to functions of several variables in the twelfth of his

famous problems at the International Congress in 1900. The first tentative steps in this di-

rection were taken by Hilbert himself and his students Blumenthal and Hecke in their study

of what are now called Hilbert (or Hilbert-Blumenthal) modular varieties. As the theory

of complex functions of several variables matured, other quotients of bounded symmet-

ric domains by arithmetic groups were studied (Siegel, Braun, and others). However, the

modern theory of Shimura varieties1 only really began with the development of the theory

of abelian varieties with complex multiplication by Shimura, Taniyama, and Weil in the

mid-1950s, and with the subsequent proof by Shimura of the existence of canonical mod-

els for certain families of Shimura varieties. In two fundamental articles, Deligne recast

the theory in the language of abstract reductive groups and extended Shimuras results on

canonical models. Langlands made Shimura varieties a central part of his program, both as

a source of representations of galois groups and as tests for the conjecture that all motivic

L-functions are automorphic. These notes are an introduction to the theory of Shimura

varieties from the point of view of Deligne and Langlands. Because of their brevity, many

proofs have been omitted or only sketched.

Notations and conventions

Unless indicated otherwise, vector spaces are assumed to be finite dimensional and free

Z-modules are assumed to be of finite rank. The linear dual Hom( V , k )of a vector space(or module)V is denotedV_. For a k -vector spaceV and ak -algebraR, V (R) denotesR k V (and similarly for Z-modules). By a lattice in an R-vector spaceV, I mean a fulllattice, i.e., a Z-submodule generated by a basis forV. The algebraic closure of a field k isdenotedk al.

A superscriptC (resp. ) denotes a connected component relative to a real topology(resp. a zariski topology). For an algebraic group, we take the identity connected compo-

nent. For example, (On)D SOn, (GLn)D GLn, and GLn(R)C consists of the nn

matrices with det > 0. For an algebraic group G overQ, G(Q)CD G(Q)\G (R)C.Following Bourbaki, I require compact topological spaces to be separated.

Semisimple and reductive groups, whether algebraic or Lie, are required to be con-nected. A simple algebraic or Lie group is a semisimple group with no connected proper

normal subgroups other than 1 (some authors say almost-simple). For a torusT, X(T )denotes the character group ofT. The inner automorphism defined by an element g is de-

noted ad(g). The derived group of a reductive group G is denotedG der (it is a semisimple

group). For more notations concerning reductive groups, see p51. For a finite extension of

fieldsL Fof characteristic zero, the torus overFobtained by restriction of scalars fromGmoverL is denoted2 (Gm)L=F.

1The term Shimura variety was introduced by Langlands (1976, 1977), although earlier Shimura

curve had been used for the varieties of dimension one (Ihara 1968).2

Thus,(Gm)L=Fhas character groupX((Gm)L=F) D ZHom(L,Fal)

(free Z-module on Hom(L,Fal

)with


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Throughout, I use the notations standard in algebraic geometry, which sometimes con-

flict with those used in other areas. For example, ifG andG 0 are algebraic groups over afield k , then a homomorphism G! G0 means a homomorphism defined over k ; ifK isa field containingk , then GK is the algebraic group over Kobtained by extension of thebase field andG(K)is the group of points ofG with coordinates inK. IfW k ,! K is ahomomorphism of fields andVis an algebraic variety (or other algebro-geometric object)

overk , then Vhas its only possible meaning: apply to the coefficients of the equations

defining V.

Let A and Bbe sets and let be an equivalence relation on A. If there exists a canonicalsurjectionA ! B whose fibres are the equivalence classes, then I say that B classifies theelements ofA modulo or that it classifies the -classes of elements ofA.

A functorFWA! Bis fully faithful if the maps HomA(a,a0)! HomB(Fa,Fa0)arebijective. The essential image of such a functor is the full subcategory ofB whose objects

are isomorphic to an object of the form F a. Thus, a fully faithful functorFWA!

Bis an

equivalence if and only if its essential image isB (Mac Lane 1998, p93).

References

In addition to those listed at the end, I refer to the following of my course notes (available

at www.jmilne.org/math/).

AG:Algebraic Geometry, v4.0, October 30, 2003.

ANT:Algebraic Number Theory, v2.1, August 31, 1998.

CFT:Class Field Theory, v3.1, May 6, 1997.

FT:Fields and galois Theory, v3.0, August 31, 2003.

MF:Modular Functions and Modular Forms, v1.1, May 22, 1997.

Prerequisites

Beyond the mathematics that students usually acquire by the end of their first year of grad-

uate work (a little complex analysis, topology, algebra, differential geometry,...), I assume

familiarity with some algebraic number theory, algebraic geometry, algebraic groups, and

elliptic modular curves.

Acknowledgements

I thank the Clay Mathematical Institute and the organizers for giving me the opportunity

to lecture on Shimura varieties, the Fields Institute for providing a excellent setting for the

Summer School, and the audience for its lively participation. Also, I thank Lizhen Ji and

Gopal Prasad for their help with references, and F. Hormann and others for alerting me to

errors in earlier versions.

the natural action of Gal(Fal

=F )), and its points in an F-algebraR are(Gm)L=F(R) D (L FR).


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8 1 HERMITIAN SYMMETRIC DOMAINS

1 Hermitian symmetric domains

In this section, I describe the complex manifolds that play the role in higher dimensions of

the complex upper half plane, or, equivalently, the open unit disk:

fz2 C j =(z) >0g D H1z! zi

zCi>

0g.Note that the map ZD (zij )7! (zij )ji identifiesHg with an open subset ofCg(gC1)=2.The symplectic group Sp2g(R) is the group fixing the alternating form

PgiD1xiyi

10

This was proved by E. Cartan, and extended to all riemannian manifolds by Myers and Steenrod.


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Example: Bounded symmetric domains. 13

PgiD1xi yi :

Sp2g(R)

D A B

C D

AtCD CtA AtD CtBD IgD

t

A Bt

CD Ig Bt

DD Dt

B .

The group Sp2g(R)acts transitively onHg by

A B

C D

ZD (AZ C B)(CZ C D)1.

The matrix

0 IgIg 0

acts as an involution on Hg, and hasiIg as its only fixed point. Thus,

Hgis homogeneous and symmetric as a complex manifold, and we shall see in (1.4) below

thatHg is in fact a hermitian symmetric domain.

Example: Bounded symmetric domains.

A domain D inCn is a nonempty open connected subset. It is symmetric if the groupHol(D) of holomorphic automorphisms ofD (as a complex manifold) acts transitively and

for some point there exists a holomorphic symmetry. For example, H1 is a symmetric

domain andD1 is a bounded symmetric domain.

THEOREM 1.3. Every bounded domain has a canonical hermitian metric (called11 the

Bergman(n) metric). Moreover, this metric has negative curvature.

PROOF( SKETCH): Initially, let Dbe any domain inCn

. The holomorphic square-integrablefunctions fW D! Cform a Hilbert space H(D) with inner product(fjg)D R

Dfgdv.

There is a unique12 (Bergman kernel) functionKW D D! C such that(a) the functionz7! K(z, )lies inH(D)for each ,(b) K(z, ) D K(, z), and11After Stefan Bergmann. When he moved to the United States in 1939, he dropped the second n from his

name.12When one ignores convergence questions, the proof is easy. Letk be a second function satisfying the

three conditions. Then

k(z, )

D Z K(z, t)k(t, )dv(t )

DZ

k(,t)K(z,t)dv(t )

D K(z, ),

which proves the uniqueness. Let

K(z, ) DPmem(z) em().Then clearlyK(z, ) D K(, z), and

fDPm(fjem)emDZ

K(, )f()dv()

(actual equality, not almost-everywhere equality, because the functions are holomorphic).


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(c) f(z) D RK(z, )f()dv()for all f2 H(D).For example, for any complete orthonormal set(em)m2N inH(D),K(z, )D

Pmem(z)

em()is such a function. IfD is bounded, then all polynomial functions on D are square-

integrable, and so certainly K(z, z) > 0 for all z. Moreover, log(K(z, z))is smooth andthe equations

h DPhij dz id zj , hij (z) D @2

@zi@zj log K(z,z),

define a hermitian metric on D , which can be shown to have negative curvature (Helgason

1978, VIII 3.3, 7.1; Krantz 1982, 1.4).

The Bergman metric, being truly canonical, is invariant under the action Hol(D). Hence,

a bounded symmetric domain becomes a hermitian symmetric domain for the Bergman

metric. Conversely, it is known that every hermitian symmetric domain can be embedded

into some Cn as a bounded symmetric domain. Therefore, a hermitian symmetric domainDhas a unique hermitian metric that maps to the Bergman metric under every isomorphism

ofD with a bounded symmetric domain. On each irreducible factor, it is a multiple of the

original metric.

EXAMPLE1.4. LetDgbe the set of symmetric complex matrices such that IgZtZis pos-itive definite. Note that(zij)7! (zij)ji identifiesDg as a bounded domain in Cg(gC1)=2.The mapZ7!(Z iIg)(ZC iIg)1 is an isomorphism ofHg ontoDg. Therefore,Dgis symmetric and Hg has an invariant hermitian metric: they are both hermitian symmetric

domains.

Automorphisms of a hermitian symmetric domain

LEMMA 1.5. Let(M, g)be a symmetric space, and letp2M. Then the subgroupKp ofIs(M,g)C fixingp is compact, and

a Kp7! a pW Is(M,g)C=Kp! M

is an isomorphism of smooth manifolds. In particular,Is(M,g)C acts transitively onM.

PROOF. For any riemannian manifold (M,g), the compact-open topology makes Is(M,g)into a locally compact group for which the stabilizer K0p of a pointp is compact (Helgason1978, IV 2.5). The Lie group structure on Is(M,g) noted above is the unique such structure

compatible with the compact-open topology (ibid. II 2.6). An elementary argument (e.g.,

MF 1.2) now shows that Is(M,g)=K0p ! M is a homeomorphism, and it follows thatthe map a 7! apW Is(M,g)! M is open. Write Is(M,g) as a finite disjoint unionIs(M,g)D FiIs(M,g)Cai of cosets of Is(M,g)C. For any two cosets the open setsIs(M,g)Cai pand Is(M,g)Caj pare either disjoint or equal, but, as Mis connected, theymust all be equal, which shows that Is(M,g)C acts transitively. Now Is(M,g)C=Kp!M is a homeomorphism, and it follows that it is a diffeomorphism (Helgason 1978, II

4.3a).


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The homomorphismupW U1! Hol(D) 15

PROPOSITION1.6. Let(M,g)be a hermitian symmetric domain. The inclusions

Is(M1, g) Is(M,g) Hol(M )

give equalities:

Is(M1, g)CD Is(M,g)CD Hol(M )C.Therefore, Hol(M )C acts transitively onM, andHol(M )C=KpD M1.

PROOF. The first equality is proved in Helgason 1978, VIII 4.3, and the second can be

proved similarly. The rest of the statement follows from (1.5).

LetH be a connected real Lie group. There need not be an algebraic groupG over Rsuch that13 G(R)CD H. However, ifH has a faithful finite-dimensional representationH ,

!GL(V ), then there exists an algebraic group G

GL(V )such that Lie(G)

D[h, h]

(insidegl(V )) wherehD Lie(H )(Borel 1991, 7.9). IfH, in addition, is semisimple, then[h, h]Dhand so Lie(G)Dhand G (R)CD H (inside GL(V )). This observation appliesto any connected adjoint Lie group and, in particular, to Hol(M )C, because the adjointrepresentation on the Lie algebra is faithful.

PROPOSITION1.7. Let(M,g) be a hermitian symmetric domain, and leth D Lie(Hol(M )C).There is a unique connected algebraic subgroupG ofGL(h)such that

G(R)CD Hol(M )C (inside GL(h)).

For such aG,G(R)CD G(R) \ Hol(M ) (inside GL(h));

thereforeG(R)C is the stablizer inG(R)ofM.

PROOF. The first statement was proved above, and the second follows from Satake 1980,

8.5.

EXAMPLE1.8. The mapz7! z1 is an antiholomorphic isometry ofH1, and every isom-etry ofH1 is either holomorphic or differs from z7! z1 by a holomorphic isometry. Inthis case,GD PGL2, and PGL2(R)acts holomorphically on CR with PGL2(R)Cas thestabilizer of

H1.

The homomorphismupW U1! Hol(D)LetU1D fz2 C j jzj D 1g (the circle group).

THEOREM 1.9. LetD be a hermitian symmetric domain. For eachp2 D, there existsa unique homomorphism upW U1! Hol(D) such thatup(z) fixes p and acts on TpD asmultiplication byz .

13For example, the (topological) fundamental group of SL2(R) is Z, and so SL2(R) has many proper

covering groups (even of finite degree). None of them is algebraic.


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EXAMPLE1.10. Let pD i2 H1, and let hWC! SL2(R) be the homomorphism zD aCib7! a bb a . Thenh(z)acts on the tangent spaceTiH1as multiplication byz=z, because

ddz

azCb

bz

Ca j

i

D a2Cb2(a

bi)2

. Forz

2U1, choose a square root

pz

2U1, and setu(z)

Dh(

pz)

mod I. Thenu(z)is independent of the choice ofpzbecauseh(1) D I. Therefore,u is a well-defined homomorphism U1 ! PSL2(R) such that u(z) acts on the tangentspaceTiH1 as multiplication byz .

Because of the importance of the theorem, I sketch a proof.

PROPOSITION1.11. Let(M, g)be symmetric space. The symmetry sp atp acts as 1onTpM, and, for any geodesic with (0)D p, sp((t))D (t ). Moreover, (M,g) is(geodesically) complete.

PROOF. Becauses2pD1,(dsp)2 D1, and sodsp acts semisimply on TpM with eigenval-ues1. Recall that for any tangent vector X atp , there is a unique geodesic W I! Mwith (0) D p,P(0) D X. If(dsp)(X) D X, thensp is a geodesic sharing these prop-erties, and so p is not an isolated fixed point ofsp. This proves that only 1occurs as aneigenvalue. If(dsp)(X)D X, thensp andt7! (t)are geodesics throughp withvelocity X, and so are equal. For the final statement, see Boothby 1975, VII 8.4.

By a canonical tensor on a symmetric space (M, g), I mean any tensor canonically

derived fromg, and hence fixed by any isometry of(M,g).

PROPOSITION1.12. On a symmetric space (M, g)every canonicalr -tensor withr odd is

zero. In particular, parallel translation of two-dimensional subspaces does not change the

sectional curvature.

PROOF. Lettbe a canonicalr -tensor. Then

tpD tp (dsp)r 1.11D (1)r tp,and sotD 0ifr is odd. For the second statement, let rbe the riemannian connection, andletR be the corresponding curvature tensor (Boothby 1975, VII 3.2, 4.4). ThenrRis anodd tensor, and so is zero. This implies that parallel translation of2-dimensional subspaces

does not change the sectional curvature.

PROPOSITION 1.13. Let(M,g) and (M0, g0) be riemannian manifolds in which paral-lel translation of2-dimensional subspaces does not change the sectional curvature. LetaW TpM! Tp0M0 be a linear isometry such that K(p,E)D K(p0,aE) for every 2-dimensional subspaceE TpM. Thenexpp(X ) 7! expp0(aX)is an isometry of a neigh-bourhood ofp onto a neighbourhood ofp0.

PROOF. This follows from comparing the expansions of the riemann metrics in terms of

normal geodesic coordinates. See Wolf 1984, 2.3.7.

PROPOSITION 1.14. If in (1.13) M and M0 are complete, connected, and simply con-nected, then there is a unique isometry W M! M0 such that(p) D p0 andd pD a.

PROOF. See Wolf 1984, 2.3.12.


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Cartan involutions 17

I now complete the sketch of the proof of Theorem 1.9. Each z with jzj D1 defines anautomorphism of(TpD, gp), and one checks that it preserves sectional curvatures. Accord-

ing to (1.11, 1.12, 1.14), there exists a unique isometry up(z)W D! D such thatdup(z)pis multiplication by z . It is holomorphic because it is C-linear on the tangent spaces. Theisometryup(z) up(z0) fixes p and acts as multiplication by z z0 on TpD, and so equalsup(zz

0).

Cartan involutions

LetG be a connected algebraic group over R, and letg7!g denote complex conjugationonG (C). An involution ofG (as an algebraic group over R) is said to beCartanif thegroup

G()(R) df

D fg

2G(C)

jg

D(g)

g (9)

is compact.

EXAMPLE 1.15. LetGD SL2, and let D ad

0 11 0

. For

a bc d

2 SL2(C), we have

a bc d

D 0 11 0 a bc d 0 11 0 1 D

dcb a

.

Thus,

SL()2 (R) D n

a bc d

2 SL2(C) j dD a,cD boD a bb a 2 GL2(C) j jaj2 C jbj2 D 1 D SU2,which is compact, being a closed bounded set in C2. Thus is a Cartan involution for SL2.

THEOREM 1.16. There exists a Cartan involution if and only ifG is reductive, in which

case any two are conjugate by an element ofG(R).

PROOF. See Satake 1980, I 4.3.

EXAMPLE 1.17. LetG be a connected algebraic group over R.

(a) The identity map on G is a Cartan involution if and only ifG(R)is compact.(b) LetGD GL(V )withVa real vector space. The choice of a basis for V determines

a transpose operatorM7! Mt, andM7!(Mt)1 is obviously a Cartan involution. Thetheorem says that all Cartan involutions ofG arise in this way.

(c) Let G ,! GL(V ) be a faithful representation ofG. ThenG is reductive if andonly ifG is stable under g7! gt for a suitable choice of a basis for V, in which case therestriction ofg7! (gt)1 toG is a Cartan involution; all Cartan involutions ofG arise inthis way from the choice of a basis for V(Satake 1980, I 4.4).

(d) Let be an involution ofG. There is a unique real formG() ofGC such that

complex conjugation onG()(C)isg7! (g). Then,G()(R)satisfies (9), and we see thatthe Cartan involutions ofG correspond to the compact forms ofG

C.


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PROPOSITION 1.18. LetG be a connected algebraic group overR. IfG (R) is compact,then every finite-dimensional real representation ofG! GL(V ) carries a G-invariantpositive definite symmetric bilinear form; conversely, if one faithful finite-dimensional real

representation ofG carries such a form, thenG(R)is compact.PROOF. Let W G! GL(V ) be a real representation ofG . IfG(R) is compact, then itsimageH in GL(V )is compact. Letdhbe the Haar measure on H, and choose a positive

definite symmetric bilinear form h j i onV. Then the form

hujvi0DZ

H

hhujhvidh

is G-invariant, and it is still symmetric, positive definite, and bilinear. For the converse,

choose an orthonormal basis for the form. Then G(R)becomes identified with a closed setof real matricesA such thatAt

A

DI, which is bounded.

REMARK 1.19. The proposition can be restated for complex representations: ifG(R) iscompact then every finite-dimensional complex representation ofG carries a G-invariant

positive definite Hermitian form; conversely, if some faithful finite-dimensional complex

representation ofG carries aG -invariant positive definite Hermitian form, then G is com-

pact. (In this case,G (R)is a subgroup of a unitary group instead of an orthogonal group.For a sesquilinear form ' to be G -invariant means that '(gu, gv)D '(u, v), g2 G(C),u, v2 V.)

Let G be a real algebraic group, and let Cbe an element ofG(R) whose square iscentral (so that adCis an involution). AC-polarizationon a real representationV ofG is

aG-invariant bilinear form ' such that the form 'C,

(u, v) 7! '(u,Cv),is symmetric and positive definite.

PROPOSITION1.20. IfadCis a Cartan involution ofG, then every finite-dimensional real

representation ofG carries aC-polarization; conversely, if one faithful finite-dimensional

real representation ofG carries aC-polarization, thenadCis a Cartan involution.

PROOF. An R-bilinear form ' on a real vector space Vdefines a sesquilinear form '0 onV (C

),

' 0W V (C) V (C) ! C, '0(u, v) D 'C(u, v).Moreover, '0is hermitian (and positive definite) if and only if'is symmetric (and positivedefinite).

Let W G! GL(V )be a real representation ofG . For anyG-invariant bilinear form 'onV, 'C is G(C)-invariant, and so

'0(gu, gv) D '0(u, v), allg2 G(C), u, v2 V (C). (10)On replacing vwithCvin this equality, we find that

' 0(gu,C (C1

gC )v) D ' 0(u,Cv), allg2 G(C), u, v2 V (C), (11)


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20/149


(a) only the characters z , 1, z1 occur in the representation ofU1 on Lie(G)C definedbyup;

(b) ad(up(1))is a Cartan involution;(c) up(1)does not project to1 in any simple factor ofG .

Conversely, letG be a real adjoint algebraic group, and letuW U1! Gsatisfy (a), (b),and (c). Then the setD of conjugates ofuby elements ofG(R)C has a natural structure ofa hermitian symmetric domain for which G (R)CD Hol(D)C andu(1)is the symmetryatu (regarded as a point ofD).

PROOF( SKETCH): LetD be a hermitian symmetric domain, and let G be the associated

group (1.7). ThenG(R)C=KpDD whereKp is the group fixingp (see 1.6). Forz2U1,up(z) acts on the R-vector space

Lie(G)=Lie(Kp) D TpD

as multiplication byz , and it acts on Lie(Kp)trivially. From this, (a) follows.The symmetrysp atp andup(1)both fixpand act as 1onTpD(see 1.11); they are

therefore equal (1.14). It is known that the symmetry at a point of a symmetric space gives

a Cartan involution ofGif and only if the space has negative curvature (see Helgason 1978,

V 2; the real form ofGdefined by adspis that attached to the compact dual of the symmetric

space). Thus (b) holds.

Finally, if the projection ofu(1)into a simple factor ofG were trivial, then that factorwould be compact (by (b); see 1.17a), and D would have an irreducible factor of compact

type.

For the converse, let D be the set ofG (R)C-conjugates ofu. The centralizerKu ofuin G(

R)C

is contained inf

g2

G(C

)j

gD

u(

1)

g

u(

1)

1

g, which, according to

(b), is compact. AsKu is closed, it also is compact. The equalityDD

G(R)C=Ku u

endowsD with the structure of smooth (even real-analytic) manifold. For this structure,

the tangent space to D at u,

TuDD Lie(G)=Lie(Ku),which, because of (a), can be identified with the subspace of Lie(G)Con which u(z) acts as

z(see (12)). This endows TuDwith aC-vector space structure for which u(z),z2 U1, actsas multiplication byz. BecauseD is homogeneous, this gives it the structure of an almost-

complex manifold, which can be shown to integrable (Wolf 1984, 8.7.9). The action of

Ku onD defines an action of it on TuD. BecauseKu is compact, there is a Ku-invariant

positive definite form onTuD(see 1.18), and becauseJD u(i)2Ku, any such form willhave the hermitian property (7). Choose one, and use the homogeneity ofD to move it

to each tangent space. This will makeD into a hermitian symmetric space, which will be

a hermitian symmetric domain because each simple factor of its automorphism group is a

noncompact semisimple group (because of (b,c)).

COROLLARY 1.22. There is a natural one-to-one correspondence between isomorphism

classes of pointed hermitian symmetric domains and pairs (G,u) consisting of a real ad-

joint Lie group and a nontrivial homomorphismuW U1! G(R)satisfying (a), (b), (c).EXAMPLE1.23. LetuW U1! PSL2(R)be as in (1.10). Then u(1) D

0 11 0

and we saw

in 1.15 that adu(

1)is a Cartan involution of SL2

, hence also of PSL2

.


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Classification of hermitian symmetric domains in terms of dynkin diagrams 21

Classification of hermitian symmetric domains in terms of dynkin dia-

grams

LetG be a simple adjoint group over R, and letube a homomorphismU1! Gsatisfying(a) and (b) of Theorem 1.21. By base extension, we get an adjoint group GC, which issimple because it is an inner form of its compact form,15 and a cocharacter DuC ofGCsatisfying the following condition:

(*) in the action ofGm on Lie(GC) defined by ad , only the charactersz , 1 , z1 occur.

PROPOSITION 1.24. The map (G,u)7! (GC, uC) defines a bijection between the sets ofisomorphism classes of pairs consisting of

(a) a simple adjoint group overR and a conjugacy class of uW U1 ! H satisfying(1.21a,b), and

(b) a simple adjoint group overC and a conjugacy class of cocharacters satisfying (*).

PROOF. Let (G, ) be as in (b), and let g 7! g denote complex conjugation on G(C)relative to the unique compact real form ofG (cf. 1.16). There is a real formH ofG such

that complex conjugation onH (C) D G(C)isg7! (1) g (1)1, andu Ddf jU1takes values inH (R). The pair(H,u)is as in (a), and the map (G, ) ! (H,u)is inverseto(H,u) 7! (HC, uC)on isomorphism classes.

Let Gbe a simple algebraic groupC. Choose a maximal torus T in Gand a base (i)i2Ifor the roots ofG relative to T. Recall, that the nodes of the dynkin diagram of(G, T )

are indexed byI. Recall also (Bourbaki 1981, VI 1.8) that there is a unique (highest) root

Q D Pnii such that, for any other rootPmii, ni mi alli . An i (or the associatednode) is said to bespecialifniD 1.

LetMbe a conjugacy class of nontrivial cocharacters ofG satisfying (*). Because all

maximal tori ofG are conjugate,Mhas a representative in X(T )X(G), and becausethe Weyl group acts simply transitively on the Weyl chambers (Humphreys 1972, 10.3)

there is a unique representative for Msuch that hi,i 0 for alli2I. The condition(*) is that16 h,i 2 f1, 0,1g for all roots . Since is nontrivial, not all the valuesh,i can be zero, and so this condition implies thathi ,i D 1 for exactly one i 2 I,which must in fact be special (otherwiseh Q ,i > 1). Thus, the Msatisfying (*) are inone-to-one correspondence with the special nodes of the dynkin diagram. In conclusion:

THEOREM1.25. The isomorphism classes of irreducible hermitian symmetric domains are

classified by the special nodes on connected dynkin diagrams.

The special nodes can be read off from the list of dynkin diagrams in, for example,

Helgason 1978, p477. In the following table, we list the number of special nodes for each

type:

Type An Bn Cn Dn E6 E7 E8 F4 G2n 1 1 3 2 1 0 0 0

15IfGC is not simple, say, GCD G1 G2, then G D ResC=R(G1) and any inner form ofG is also therestriction of scalars of a C-group; but such a group can not be compact (look at a subtorus).

16

The with this property are sometimes said to be minuscule(cf. Bourbaki 1981, pp226227).


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In particular, there are no irreducible hermitian symmetric domains of type E8,F4, orG2and, up to isomorphism, there are exactly2of typeE6and1of typeE7. It should be noted

that not every simple real algebraic group arises as the automorphism group of a hermitian

symmetric domain. For example, PGLnarises in this way only for n D 2.NOTES. For introductions to smooth manifolds and riemannian manifolds, see Boothby

1975 and Lee 1997. The ultimate source for hermitian symmetric domains is Helgason

1978, but Wolf 1984 is also very useful, and Borel 1998 gives a succinct treatment close

to that of the pioneers. The present account has been influenced by Deligne 1973a and

Deligne 1979.


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23

2 Hodge structures and their classifying spaces

We describe various objects and their parameter spaces. Our goal is a description of hermi-

tian symmetric domains as the parameter spaces for certain special hodge structures.

Reductive groups and tensors

LetG be a reductive group over a fieldk of characteristic zero, and let W G! GL(V )bea representation ofG. Thecontragredientordual_ofis the representation ofG on thedual vector spaceV_defined by

(_(g) f)(v) D f((g1) v), g2 G, f2 V_, v2 V .

A representation is said to be self-dualif it is isomorphic to its contragredient.

Anr -tensorofV is a multilinear map

tW V V! k (r -copies ofV ).

For anr -tensort, the condition

t (gv1, . . . , gvr ) D (v1, . . . , vr ), all vi2 V ,

ong defines a closed subgroup of GL(V )tof GL(V ). For example, iftis a nondegenerate

symmetric bilinear formV V! k, then GL(V )tis the orthogonal group. For a set Tof tensors ofV,Tt2T

GL(V )tis called thesubgroup ofGL(V )fixing thet

2T.

PROPOSITION2.1. For any faithful self-dual representation G! GL(V ) ofG, there existsa finite setTof tensors ofVsuch thatG is the subgroup ofGL(V )fixing thet2 T.PROOF. In Deligne 1982, 3.1, it is shown there exists a possibly infinite set T with this

property, but, because G is noetherian as a topological space (i.e., it has the descending

chain condition on closed subsets), a finite subset will suffice.

PROPOSITION2.2. LetG be the subgroup ofGL(V )fixing the tensorst2 T. Then

Lie(G) D

g2 End(V )Pj

t (v1, . . . , gvj , . . . , vr ) D 0, allt2 T, vi2 V

.

PROOF. The Lie algebra of an algebraic group G can be defined to be the kernel of

G(k["])! G(k). Here k["] is the k-algebra with "2 D 0. Thus Lie(G) consists ofthe endomorphisms1 C g"ofV (k["])such that

t((1 C g")v1, (1 C g")v2, . . . ) D t (v1, v2, . . . ) , allt2 T, vi2 V .

On expanding this and cancelling, we obtain the assertion.

Flag varieties

Fix a vector spaceVof dimensionn over a fieldk .


24/149

24 2 HODGE STRUCTURES AND THEIR CLASSIFYING SPACES

The projective space P(V )

The set P(V )of one-dimensional subspacesL ofVhas a natural structure of an algebraicvariety: the choice of a basis forVdetermines a bijection P(V )

!Pn1, and the structure

of an algebraic variety inherited by P(V )from the bijection is independent of the choice ofthe basis.

Grassmann varieties

LetGd(V ) be the set ofd-dimensional subspaces ofV, some 0 < d < n. Fix a basis

for V. The choice of a basis for W then determines a d n matrix A(W ) whose rowsare the coordinates of the basis elements. Changing the basis for W multiplies A(W) on

the left by an invertible d dmatrix. Thus, the family of minors of degreed ofA(W )is well-determined up to multiplication by a nonzero constant, and so determines a point

P ( W ) in P(nd)1. The mapW 7! P ( W )W Gd(V )! P(

nd)1 identifiesGd(V ) with a

closed subvariety ofP(nd)1 (AG, 5.38). A coordinate-free description of this map is given

by

W7!VdWW Gd(V ) ! P(VdV ). (13)Let Sbe a subspace ofV of complementary dimension nd, and let Gd(V )Sbe the set

ofW2 Gd(V ) such that W\SD f0g. Fix a W02 Gd(V )S, so that VD W0 S. For anyW2 Gd(V )S, the projectionW! W0 given by this decomposition is an isomorphism,and soWis the graph of a homomorphism W0! S:

w7! s () (w, s) 2 W .Conversely, the graph of any homomorphismW0! S lies inGd(V )S. Thus,

Gd(V )SD Hom(W0,S). (14)

When we regard Gd(V )Sas an open subvariety ofGd(V ), this isomorphism identifies it

with the affine space A(Hom(W0,S)) defined by the vector space Hom(W0, S). Thus,Gd(V )is smooth, and the tangent space toGd(V )atW0,

TW0(Gd(V ))D

Hom(W0, S) D

Hom(W0, V=W0). (15)

Flag varieties

The above discussion extends easily to chains of subspaces. LetdD (d1, . . . , d r ) be asequence of integers with n >d1 > >dr >0, and letGd(V )be the set of flags

FW V V1 Vr 0 (16)

withVi a subspace ofVof dimensiondi . The map

Gd(V )

F7!(Vi)! Qi Gdi(V ) QiP(V

di

V )


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Hodge structures 25

realizes Gd(V ) as a closed subset ofQ

i Gdi(V ) (Humphreys 1978, 1.8), and so it is a

projective variety. The tangent space to Gd(V ) at the flag F consists of the families of

homomorphisms

'i

W Vi

! V=Vi

, 1 i r, (17)satisfying the compatibility condition

'ijViC1 'iC1 mod ViC1.ASIDE2.3. A basis e1, . . . , enfor V isadapted tothe flag Fif it contains a basis e1, . . . , ejifor eachVi. Clearly, every flag admits such a basis, and the basis then determines the flag.

Because GL(V ) acts transitively on the set of bases for V, it acts transitively on Gd(V ).

For a flagF, the subgroupP(F)stabilizingFis an algebraic subgroup of GL(V ), and the

map

g

7!gF0

WGL(V )=P (F0)

!Gd(V )

is an isomorphism of algebraic varieties. Because Gd(V ) is projective, this shows that

P (F0)is a parabolic subgroup of GL(V ).

Hodge structures

Definition

For a real vector space V, complex conjugation onV (C) DdfC RV is defined byz vD z v.

An R-basise1, . . . , emfor Vis also a C-basis forV (C)andP

ai eiDPaiei .Ahodge decompositionof a real vector spaceVis a decomposition

V (C) DM

p,q2ZZVp,q

such thatVq,p is the complex conjugate ofVp,q . Ahodge structureis a real vector space

together with a hodge decomposition. The set of pairs (p,q) for which Vp,q 6D 0 iscalled thetypeof the hodge structure. For each n,

LpCqDn V

p,q is stable under complex

conjugation, and so is defined over R, i.e., there is a subspace Vn ofV such thatVn(C)D

LpCqDn V

p,q (see AG 14.5). ThenVDL

n Vnis called theweight decompositionofV.

IfVD

Vn

, thenVis said to have weight n.

Anintegral(resp. rational)hodge structureis a free Z-module of finite rankV (resp.Q-vector space) together with a hodge decomposition ofV (R) such that the weight decom-position is defined over Q.

EXAMPLE2.4. Let Jbe a complex structure on a real vector space V, and define V1,0 andV0,1 to be the Ciandieigenspaces ofJacting on V (C). Then V (C) D V1,0V0,1is a hodge structure of type (1, 0), (0, 1), and every real hodge structure of this typearises from a (unique) complex structure. Thus, to give a rational hodge structure of type

(1, 0), (0, 1)amounts to giving a Q-vector spaceVand a complex structure onV (R),and to give an integral hodge structure of type (1,0),(0, 1) amounts to giving aC-vectorspaceVand a lattice V (i.e., a Z-submodule generated by an R-basis forV).


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EXAMPLE 2.5. LetX be a nonsingular projective algebraic variety overC. ThenHDHn(X,Q)has a Hodge structure of weight n for whichHp,q Hn(X,C)is canonicallyisomorphic toHq(X,p)(Voisin 2002, 6.1.3).

EXAMPLE 2.6. LetQ(m) be the hodge structure of weight2m on the vector spaceQ.Thus,(Q(m))(C) D Q(m)m,m. Define Z(m)and R(m)similarly.17

The hodge filtration

Thehodge filtrationassociated with a hodge structure of weight n is

FW Fp FpC1 , Fp DLrpVr,s V (C).Note that forp C qD n,

Fq DLsqVs,r DLsqVr,s DLrpVr,sand so

Vp,q D Fp \ Fq. (18)EXAMPLE2.7. For a hodge structure of type(1,0),(0, 1), the hodge filtration is

(F1 F0 F2) D (V (C) V0,1 0).

The obviousR-linear isomorphism V! V (C)=F0 defines the complex structure on Vnoted in (2.4).

Hodge structures as representations ofS

LetS be C regarded as a torus overR. It can be identified with the closed subgroup ofGL2(R)of matrices of the form18

a bb a

. Then S(C)C C with complex conjuga-tion acting by the rule (z1, z2)D(z2, z1). We fix the isomorphism SCDGm Gm so thatS(R) ! S(C)isz7! (z,z), and we define the weight homomorphism wWGm! S so thatGm(R)

w! S(R)is r7! r1WR! C.The characters ofSC are the homomorphisms (z1, z2)7! zp1 zq2 , (r, s)2Z Z. Thus,

X(S)D

Z

Z with complex conjugation acting as (p,q)

7! (q,p), and to give a

representation ofS on a real vector space V amounts to giving aZ Z-grading ofV (C)such that Vp,q D Vq,p for all p, q (see p19). Thus, to give a representation ofS on areal vector space V is the same as to give a hodge structure on V. Following Deligne

1979, 1.1.1.1, we normalize the relation as follows: the homomorphism hWS! GL(V )corresponds to the hodge structure on Vsuch that

hC(z1, z2)vD zp1 zq2 vfor v2 Vp,q . (19)17It would be a little more canonical to take the underlying vector space of Q(m)to be(2i)mQ because

this makes certain relations invariant under a change of the choice ofiD p1in C.18This is the transpose of the matrix ofa C ib acting on C relative to the basis1, i, but it gives the correct

action on the tangent space, namely, if jzj D 1, thenh(z)acts asz2

(see 1.10).


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Hodge structures 27

In other words,

h(z)vD zpzqvfor v2 Vp,q . (20)

Note the minus signs! The associated weight decomposition has

VnD fv2 Vj wh(r )vD r ng, whD h w. (21)

Let hbe the cocharacter of GL(V )defined by

h(z) D hC(z,1). (22)

Then the elements ofFp

hV are sums ofv2 V (C) satisfying h (z) vD zrv for some

r p.To give a hodge structure on a Q-vector spaceVamounts to giving a homomorphism

hWS ! GL(V (R))such that whis defined over Q.

EXAMPLE 2.8. By definition, a complex structure on a real vector space is a homomor-

phismhWC!EndR(V )ofR-algebras. ThenhjCWC!GL(V )is a hodge structure oftype(1, 0), (0, 1)whose associated complex structure (see 2.4) is that defined by h.19

EXAMPLE 2.9. The Hodge structure Q(m) corresponds to the homomorphism hW S!GmR,h(z) D (zz)m.

The Weil operator

For a hodge structure(V , h), the R-linear mapCD h(i)is called theWeil operator. NotethatCacts asi qp onVp,q and thatC2 D h(1)acts as(1)n onVn.

EXAMPLE2.10. IfVis of type(1,0),(0, 1), thenCcoincides with theJof (2.4). Thefunctor(V , (V1,0, V0,1))7! (V , C ) is an equivalence from the category of real hodgestructures of type(1, 0), (0, 1)to the category of complex vector spaces.

Hodge structures of weight0.

LetVbe a hodge structure of weight0. ThenV0,0 is invariant under complex conjugation,

and soV0,0 D V00(C), whereV00 D V0,0 \ V(see AG 14.5). Note that

V00 D Ker(V! V (C)=F0). (23)19This partly explains the signs in (19); see also Deligne 1979, 1.1.6. Following Deligne 1973b, 8.12,

and Deligne 1979, 1.1.1.1,hC(z1, z2)vp,q D zp1 zq2 vp,q has become the standard convention in the theory

of Shimura varieties. When one identifies complex structures on a real vector space with Hodge structures

of type (1,0),(0,1) (or abelian varieties with hodge structures using H1 rather than H1), then it is more

convenient to use the conventionhC(z1, z2)vp,q D zq1 zp2 vp,q (note the switch). I tried this in the lectures, but

have abandoned it because it causes too much confusion. Following Deligne 1971a, 2.1.5.1, the convention

hC(z1, z2)vp,q

D zp

1 z

q

2vp,q

is commonly used in hodge theory (e.g., Voisin 2002, p147).


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Tensor products of hodge structures

The tensor product of hodge structuresV andW of weightm and n is a hodge structure

of weightmC

n:

V W , (V W )p,q DLrCr 0Dp,sCs0DqVr,s Vr 0,s0 .In terms of representations ofS,

( V , hV) (W,hW) D (V W , hV hW).

Morphisms of hodge structures

A morphism of Hodge structures is a linear map V! W sendingVp,q intoWp,q for allp, q. In other words, it is a morphism(V , h

V)!

(W,hW

)of representations ofS

.

Hodge tensors

LetRD Z,Q, orR, and let (V , h) be an R-hodge structure of weight n. A multilinearformtW Vr ! Ris ahodge tensorif the map

V V V! R(nr=2)

it defines is a morphism of hodge structures. In other words, tis a hodge tensor if

t(h(z)v1,h(z)v2, . . . ) D (zz)nr=2

tR(v1, v2, . . . ) , allz2 C, vi2 V (R),or if P

pi6DP

qi) tC(vp1,q11 , vp2,q22 , . . . ) D 0, vpi,qii 2 Vpi,qi . (24)Note that, for a hodge tensort,

t (Cv1, Cv2, . . . ) D t (v1, v2, . . . ) .

EXAMPLE 2.11. Let(V , h)be a hodge structure of type(1, 0), (0, 1). A bilinear formtW V V! R is a hodge tensor if and only ift(Ju, Jv) D t(u, v)for allu, v2 V.

Polarizations

Let(V , h) be a hodge structure of weight n. A polarization of(V , h) is a hodge tensor

W V V! R such that C(u, v) Ddf (u,Cv)is symmetric and positive definite. Then is symmetric or alternating according as n is even or odd, because

(v, u) D (Cv, C u ) D C(Cv, u) D C(u,Cv) D (u,C2v) D (1)n(u, v).

More generally, let (V,h) be an R-hodge structure of weight n where R is Z orQ. Apolarizationof(V , h)is a bilinear form W V V!Rsuch that R is a polarization of(V (R),h).


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Variations of hodge structures 29

EXAMPLE 2.12. Let (V,h) be an R-hodge structure of type (1, 0), (0, 1) with RDZ, Q, orR, and let JD h(i). A polarization of (V,h) is an alternating bilinear form W V V! Rsuch that, foru, v2 V (R),

R(Ju,Jv) D (u, v), andR(u,Ju) >0 ifu 6D 0.

(These conditions imply that R(u,Jv)is symmetric.)

EXAMPLE2.13. Let Xbe a nonsingular projective variety overC. The choice of an embed-dingX ,!PN determines a polarization on the primitive part ofHn(X,Q)(Voisin 2002,6.3.2).

Variations of hodge structures

Fix a real vector space V, and let S be a connected complex manifold. Suppose that,

for each s2 S, we have a hodge structure hs onV of weight n (independent ofs). LetV

p,qs D Vp,qhs andF

ps D Fps VD FphsV.

The family of hodge structures(hs)s2SonVis said to becontinuousif, for fixedpandq, the subspaceV

p,qs varies continuously withs. This means that the dimensiond(p, q)of

Vp,q

s is constant and the map

s7! Vp,qs W S! Gd(p,q)(V )is continuous.

A continuous family of hodge structures(Vp,q

s )s is said to beholomorphicif the hodgefiltrationFs varies holomorphically with s. This means that the map ',

s7! Fs W S! Gd(V )is holomorphic. Hered D (. . . ,d(p),. . .) whered(p) D dim Fps VD

Prpd(r,q). Then

the differential of' at s is a C-linear map

d'sW TsS! TFs(Gd(V ))(17) LpHom(Fps , V=Fps ).

If the image ofd's is contained inLpHom(F

ps , F

p1s =F

ps ),

for alls , then the holomorphic family is called a variation of hodge structures onS.

Now letTbe a family of tensors onVincluding a nondegenerate bilinear form t0, and

letdWZ Z ! N be a function such thatd(p,q) D 0for almost allp, q;d(q,p) D d(p,q);d(p,q) D 0unlessp C qD n.

DefineS (d , T )to be the set of all hodge structures h onVsuch that


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dim Vp,qh

D d(p,q)for allp, q; eacht2 Tis a hodge tensor forh; t0 is a polarization for h.

ThenS (d , T )acquires a topology as a subspace ofQ

d(p,q)6D0Gd(p,q)(V ).

THEOREM2.14. LetSC be a connected component ofS (d , T ).(a) If nonempty, SC has a unique complex structure for which (hs) is a holomorphic

family of hodge structures.

(b) With this complex structure, SCis a hermitian symmetric domain if(hs) is a variationof hodge structures.20

(c) Every irreducible hermitian symmetric domain is of the formSCfor a suitableV,d,andT.

PROOF( SKETCH). (a) Let SC

DS (d , T )C. Because the hodge filtration determines the

hodge decomposition (see (18)), the map x7! Fs W SC '! Gd(V ) is injective. LetG bethe smallest algebraic subgroup of GL(V )such that

h(S) G, allh 2 SC (25)

(take G to be the intersection of the algebraic subgroups of GL(V ) with this property),

and letho2 SC. For any g2 G(R)C, ghog1 2 SC, and it can be shown that the mapg7! g ho g1W G(R)C! SCis surjective:

SCD G(R)C ho.

The subgroupKoofG(R)Cfixinghois closed, and soG(R)C=Kois a smooth (in fact, realanalytic) manifold. Therefore,SC acquires the structure of a smooth manifold from

SCD (G(R)C=Ko) hoD G(R)C=Ko.

LetgD Lie(G). From S ho! G Ad! g End(V ), we obtain Hodge structures ong andEnd(V ). Clearly,g00 D Lie(Ko)and soThoSCD g=g00. In the diagram,

ThoSCD g=g00 End(V )=End(V )00

g(C)=F0(23) D

End(V (C))=F0(23) D DThoGd(V ).

(26)

the map from top-left to bottom-right is (d')ho, which therefore mapsThoSC onto a com-

plex subspace ofThoGd(V ). Since this is true for all ho2 SC, we see that ' identifiesSCwith an almost-complex submanifoldGd(V ). It can be shown that this almost-complex

20In the preliminary version, I claimed that this was if and only if, but, as Fritz Hormann pointed out to

me, the only if is not true. For example, let VD R2 with the standard alternating form. Then the functionsd(1, 0)Dd(0, 1)D1 and d(5, 0)Dd(0, 5)D1 give the same setsS (d , T )but only the first is a variationof hodge structures. Theu given naturally by the seconddis the fifth power of that given by the first d ,and

u(z)doesnt act as multiplication by z on the tangent space.


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Variations of hodge structures 31

structure is integrable, and so provides SC with a complex structure for which ' is holo-morphic. Clearly, this is the only (almost-)complex structure for which this is true.

(b) See Deligne 1979, 1.1.

(c) Given an irreducible hermitian symmetric domain D, choose a faithful self-dual rep-resentationG! GL(V )of the algebraic group G associated withD (as in 1.7). BecauseVis self-dual, there is a nondegenerate bilinear formt0on Vfixed byG . Apply Theorem

2.1 to find a set of tensors T such thatG is the subgroup of GL(V )fixing the t2 T. Letho be the composite S

z 7!z=z! U1 uo! GL(V )withuo as in (1.9). Then,ho defines a hodgestructure on V for which the t2 Tare hodge tensors and to is a polarization. One cancheck thatD is naturally identified with the component ofS (d , T )Ccontaining this hodgestructure.21

REMARK2.15. The mapSC! Gd(V )in the proof is an embedding of smooth manifolds(injective smooth map that is injective on tangent spaces and maps SC homeomorphicallyonto its image). Therefore, if a smooth map T! Gd(V )factors into

T ! SC! Gd(V ),

then will be smooth. Moreover, if the mapT! Gd(V ) is defined by a holomorphicfamily of hodge structures on T, and it factors through SC, then will be holomorphic.

ASIDE 2.16. As we noted in (2.5), for a nonsingular projective varietyV overC, the co-homology group Hn(V (C),Q) has a natural Hodge structure of weight n. Now considera regular map W V! Sof nonsingular varieties whose fibres Vs (s2 S) are nonsingu-lar projective varieties of constant dimension. The vector spacesH

n

(Vs,Q) form a localsystem ofQ-vector spaces on S, and Griffiths showed that the Hodge structures on themform a variation of hodge structures in a slightly more general sense than that defined above

(Voisin 2002, Proposition 10.12).

NOTES. Theorem 2.14 is taken from Deligne 1979.

21Given a pair(V, (Vp,q )p,q , T ), defineLto be the sub-Lie-algebra of End(V )fixing thet2 T, i.e., suchthat P

i t (v1, . . . , gvi , . . . , vr ) D 0.Then L has a hodge structure of weight 0. We say that (V,(Vp,q )p,q , T ) is special if L is of type

(1,1),(0,0),(1,1). The familyS (d , T )Ccontaining(V, (Vp,q )p,q , T )is a variation of hodge structuresif and only if(H , T )is special.


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32 3 LOCALLY SYMMETRIC VARIETIES

3 Locally symmetric varieties

In this section, we study quotients of hermitian symmetric domains by certain discrete

groups.

Quotients of hermitian symmetric domains by discrete groups

PROPOSITION 3.1. LetD be a hermitian symmetric domain, and let be a discrete sub-

group ofHol(D)C. If is torsion free, then acts freely on D, and there is a uniquecomplex structure on nD for which the quotient map W D! nD is a local isomor-phism. Relative to this structure, a map ' from nD to a second complex manifold isholomorphic if and only if' is holomorphic.PROOF. Let be a discrete subgroup of Hol(D)C. According to (1.5, 1.6), the stabilizerKpof any point p2 Dis compact and g7! gpW Hol(D)C=Kp! Dis a homeomorphism,and so (MF, 2.5):

(a) for anyp2 D, fg2 j gpD pg is finite;(b) for anyp 2 D, there exists a neighbourhood U ofp such that, for g2 , gU is

disjoint fromU unlessgpD p;(c) for any pointsp, q2D not in the same -orbit, there exist neighbourhoodsU ofp

andV ofq such thatgU\ VD ; for allg2 .Assume is torsion free. Then the group in (a) is trivial, and so acts freely on D.

Endow nD with the quotient topology. IfU and V are as in (c) , then U and Vare disjoint neighbourhoods ofp and q, and so nD is separated. Letq2 nD, andlet p2

1

(q). IfU is as in (b), then the restriction of to U is a homeomorphismU! U, and it follows that nDa manifold.

Define a C-valued function fon an open subsetU ofnDto be holomorphic iff is holomorphic on 1U. The holomorphic functions form a sheaf on nDfor which isa local isomorphism of ringed spaces. Therefore, the sheaf defines a complex structure on

nDfor which is a local isomorphism of complex manifolds.Finally, let 'WnD! M be a map such that ' is holomorphic, and let f be a

holomorphic function on an open subset U ofM. Then f ' is holomorphic becausef ' is holomorphic, and so ' is holomorphic.

When is torsion free, we often write D()for

nDregarded as a complex manifold.

In this case, D is the universal covering space ofD() and is the group of covering

transformations; moreover, for any pointp ofD, the map

g7! [image under of any path fromp to gp]W! 1(D(),p)

is an isomorphism (Hatcher 2002, 1.40).

Subgroups of finite covolume

We shall only be interested in quotients ofD by big discrete subgroups of Aut(D)C.

This condition is conveniently expressed by saying that nDhas finite volume. By defini-


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Arithmetic subgroups 33

tion,D has a riemannian metricg and hence a volume element : in local coordinates

Dp

det(gij(x))dx1 ^ . . . ^ dxn.

Since g is invariant under , so also is , and so it passes to the quotient nD. Thecondition is that

RnD < 1.

For example, letDD H1 and let D PSL2(Z). Then

FD fz2 H1j jzj >1, 12 <


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THEOREM3.3. LetG be a reductive group overQ, and let be an arithmetic subgroup ofG(Q).

(a) The space nG(R)has finite volume if and only ifHom(G,Gm)D0 (in particular,nG(R)has finite volume ifG is semisimple).

24

(b) The space nG(R) is compact if and only ifHom(G,Gm)D 0andG (Q) containsno unipotent element (other than 1).

PROOF. Borel 1969, 13.2, 8.4, or Platonov and Rapinchuk 1994, Theorem 4.13, p213,

Theorem 4.12, p210. [The intuitive reason for the condition in (b) is that the rational

unipotent elements correspond to cusps (at least in the case of SL2 acting on H1), and so

no rational unipotent elements means no cusps.]

EXAMPLE 3.4. Let B be a quaternion algebra over Q such that B

QR

M2(R), andlet G be the algebraic group over Q such that G(Q) is the group of elements in B ofnorm 1. The choice of an isomorphismBQR! M2(R) determines an isomorphismG(R)! SL2(R), and hence an action ofG (R) on H1. Let be an arithmetic subgroupofG(Q).

IfB M2(Q), then G SL2, which is semisimple, and so n SL2(R) (hence alsonH1) has finite volume. However, SL2(Q)contains the unipotent element

1 10 1

, and so

n SL2(R)is not compact.IfB6 M2(Q), it is a division algebra, and so G(Q) contains no unipotent element

6D 1 (for otherwise B would contain a nilpotent element). Therefore, nG(R) (hencealso

nH1) is compact

Letk be a subfield ofC. An automorphism of ak -vector spaceVis said to be neatif its eigenvalues inC generate a torsion free subgroup ofC (which implies that doesnot have finite order). LetG be an algebraic group over Q. An elementg2G (Q)is neatif(g) is neat for one faithful representation G ,! GL(V ), in which case (g) is neat forevery representation ofG defined over a subfield ofC (apply Waterhouse 1979, 3.5). Asubgroup ofG(Q)is neatif all its elements are.

PROPOSITION 3.5. LetG be an algebraic group overQ, and let be an arithmetic sub-group ofG(Q). Then, contains a neat subgroup 0 of finite index. Moreover, 0 can

be defined by congruence conditions (i.e., for some embedding G ,! GLn and integerN, 0D fg2 j g 1 mod Ng).

PROOF. Borel 1969, 17.4.

LetHbe a connected real Lie group. A subgroup ofH isarithmeticif there exists

an algebraic groupG over Q and an arithmetic subgroup0ofG(Q)such that 0 \G(R)Cmaps onto under a surjective homomorphismG(R)C! Hwith compact kernel.

24Recall (cf. the Notations) that Hom(G,Gm)D 0 means that there is no nonzero homomorphism G!Gm defined overQ.


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Brief review of algebraic varieties 35

PROPOSITION 3.6. Let H be a semisimple real Lie group that admits a faithful finite-

dimensional representation. Every arithmetic subgroup ofH is discrete of finite covol-

ume, and it contains a torsion free subgroup of finite index.

PROOF. Let W G(R)C H and 0 G(Q) be as in the definition of arithmetic sub-group. Because Ker()is compact, is proper (Bourbaki 1989, I 10.3) and, in particular,

closed. Because 0 is discrete in G (R), there exists an openU G(R)C whose intersec-tion with 0 is exactly the kernel of0 \ G(R)C! . Now (G(R)C U )is closed inH, and its complement intersects inf1g. Therefore, is discrete in H. It has finitecovolume because 0nG(R)C maps onto nHand we can apply (3.3a). Let 1 be a neatsubgroup of0 of finite index (3.5). The image of1 in Hhas finite index in , and its

image under any faithful representation ofHis torsion free.

REMARK 3.7. There are many nonarithmetic discrete subgroup in SL2(R) of finite co-

volume. According to the Riemann mapping theorem, every compact riemann surface ofgenus g 2 is the quotient ofH1 by a discrete subgroup of PGL2(R)C acting freely onH1.Since there are continuous families of such riemann surfaces, this shows that there are

uncountably many discrete cocompact subgroups in PGL2(R)C(therefore also in SL2(R)),but there only countably many arithmetic subgroups.

The following (Fields medal) theorem of Margulis shows that SL2 is exceptional in

this regard: let be a discrete subgroup of finite covolume in a noncompact simple real

Lie group H; then is arithmetic unless H is isogenous to SO(1, n) or SU(1, n) (see

Witte 2001, 6.21 for a discussion of the theorem). Note that, because SL2(R)is isogenousto SO(1, 2), the theorem doesnt apply to it.

Brief review of algebraic varieties

Let k be a field. An affine k-algebra is a finitely generated k-algebra A such that Ak kal isreduced (i.e., has no nilpotents). Such an algebra is itself reduced, and when k is perfect every

reduced finitely generatedk-algebra is affine.

LetA be an affinek -algebra. Define specm(A)to be the set of maximal ideals in A endowed

with the topology having as basis D(f),D(f)D fmj f =2 mg, f2 A. There is a unique sheafofk-algebras Oon specm(A)such that O(D(f)) D Af for all f. HereAf is the algebra obtainedfromAby inverting f. Any ringed space isomorphic to a ringed space of the form

Specm(A)

D(specm(A),O)

is called anaffine varietyoverk . The stalk atm is the local ringAm, and so Specm(A)is a locally

ringed space.

This all becomes much more familiar when k is algebraically closed. When we writeADk[X1, . . . , X n]=a, the space specm(A)becomes identified with the zero set ofaink

n endowed with

the zariski topology, and O becomes identified with the sheaf ofk -valued functions on specm(A)

locally defined by polynomials.

A topological spaceVwith a sheaf ofk-algebras Ois aprevarietyoverk if there exists a finite

covering(Ui) ofV by open subsets such that (Ui ,OjUi) is an affine variety over k for all i . Amorphism of prevarieties overk is simply a morphism of ringed spaces ofk-algebras. A prevariety

V overk isseparatedif, for all pairs of morphisms ofk-prevarieties,W Z V, the subset ofZon which and agree is closed. A variety overk is a separated prevariety overk .


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Alternatively, the varieties overk are precisely the ringed spaces obtained from geometrically-

reduced separated schemes of finite type overk by deleting the nonclosed points.

A morphism of algebraic varieties is also called a regular map, and the elements ofO(U )are

called theregular functionson U.For the variety approach to algebraic geometry, see AG, and for the scheme approach, see

Hartshorne 1977.

Algebraic varieties versus complex manifolds

The functor from nonsingular algebraic varieties to complex manifolds

For a nonsingular variety V overC, V (C) has a natural structure as a complex manifold.More precisely:

PROPOSITION 3.8. There is a unique functor(V ,OV)7! (Van,OVan ) from nonsingularvarieties overC to complex manifolds with the following properties:

(a) as sets, VD Van, every zariski-open subset is open for the complex topology, andevery regular function is holomorphic;25

(b) ifVD An, thenVan D Cn with its natural structure as a complex manifold;(c) if'W V! W isetale, then 'anW Van ! Wan is a local isomorphism.

PROOF. A regular map 'W V! W is etale if the map d'pW TpV! TpW is an isomor-phism for all p

2V. Note that conditions (a,b,c) determine the complex-manifold structure

on any open subvariety ofAn and also on any varietyVthat admits anetale map to an opensubvariety ofAn. Since every nonsingular variety admits a zariski-open covering by such V(AG, 4.31), this shows that there exists at most one functor satisfying (a,b,c), and suggests

how to define it.

Obviously, a regular map 'W V!W is determined by 'anW Van !Wan, but not everyholomorphic map Van ! Wan is regular. For example,z7! ezWC! C is not regular.Moreover, a complex manifold need not arise from a nonsingular algebraic variety, and

two nonsingular varietiesV andW can be isomorphic as complex manifolds without being

isomorphic as algebraic varieties (Shafarevich 1994, VIII 3.2). In other words, the functor

V 7! Van is faithful, but it is neither full nor essentially surjective on objects.

REMARK3.9. The functor V 7! Van can be extended to all algebraic varieties once one hasthe notion of a complex manifold with singularities. This is called a complex space. For

holomorphic functions f1, . . . ,fr on a connected open subset U ofCn, letV (f1, . . . ,fr )denote the set of common zeros of the fi inU; one endows V (f1, . . . ,fr )with a natural

structure of ringed space, and then defines a complex space to be a ringed space (S ,OS)

that is locally isomorphic to one of this form (Shafarevich 1994, VIII 1.5).

25These conditions require that the identity map V ! V be a map of ringed spaces (Van,OVan )!(V ,OV). This map is universal.


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Algebraic varieties versus complex manifolds 37

Necessary conditions for a complex manifold to be algebraic

3.10. Here are two necessary conditions for a complex manifoldMto arise from an alge-

braic variety.

(a) It must be possible to embed M as an open submanifold of a compact complex

manfoldMin such a way that the boundary MMis a finite union of manifoldsof dimension dim M 1.

(b) IfMis compact, then the field of meromorphic functions onMmust have transcen-

dence degree dim M over C.

The necessity of (a) follows from Hironakas theorem on the resolution of singularities,

which shows that every nonsingular varietyVcan be embedded as an open subvariety of a

complete nonsingular varietyV in such a way that the boundary V Vis a divisor withnormal crossings (see p40), and the necessity of (b) follows from the fact that, when V is

complete and nonsingular, the field of meromorphic functions on Van coincides with the

field of rational functions onV(Shafarevich 1994, VIII 3.1).

Here is one positive result: the functor

fprojective nonsingular curves over Cg ! fcompact riemann surfacesgis an equivalence of categories (see MF, pp88-91, for a discussion of this theorem). Since

the proper zariski-closed subsets of algebraic curves are the finite subsets, we see that for

riemann surfaces the condition (3.10a) is also sufficient: a riemann surface Mis algebraic

if and only if it is possible to embed M in a compact riemann surface Min such a way thatthe boundary M M is finite. The maximum modulus principle (Cartan 1963, VI 4.4)shows that a holomorphic function on a connected compact riemann surface is constant.

Therefore, if a connected riemann surface M is algebraic, then every bounded holomorphicfunction on M is constant. We conclude thatH1 does not arise from an algebraic curve,

because the functionz7! zizCi is bounded, holomorphic, and nonconstant.

For any lattice in C, the Weierstrass } function and its derivative embedC=intoP2(C) (as an elliptic curve). However, for a lattice in C2, the field of meromorphicfunctions onC2= will usually have transcendence degree < 2, and soC2= is not analgebraic variety.26 For quotients ofCg by a lattice , condition (3.10b) is sufficient foralgebraicity (Mumford 1970, p35).

Projective manifolds and varieties

A complex manifold (resp. algebraic variety) is projective if it is isomorphic to a closed

submanifold (resp. closed subvariety) of a projective space. The first truly satisfying theo-

rem in the subject is the following:

THEOREM3.11 (CHOW 1949). Every projective complex manifold has a unique structure

of a nonsingular projective algebraic variety, and every holomorphic map of projective

complex manifolds is regular for these structures. (Moreover, a similar statement holds for

complex spaces.)

26A complex torus Cg=is algebraic if and only if it admits a riemann form (see 6.7 below). When isthe lattice in C2 generated by(1, 0), (i, 0), (0, 1), (,)with nonreal, C2= does not admit a riemann form

(Shafarevich 1994, VIII 1.4).


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PROOF. See Shafarevich 1994, VIII 3.1 (for the manifold case).

In other words, the functor V 7! Van is an equivalence from the category of (non-singular) projective algebraic varieties to the category of projective complex (man

Milne - Introduction to Shimura Varieties

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