Table Of ContenteMI Lecture Series in Mathematics 1
Lectures on the structure of algebraic
groups and geometric applications
eMf Lecture Series in Mathematics
Editors
Vikraman Balaji, Chennai Mathematical Institute, Chennai.
Rajeeva Karandikar, Chennai Mathematical Institute, Chennai.
Pramathanath Sastry, Chennai Mathematical Institute, Chennai.
Volumes published so far
1. Lectures on the structure of algebraic groups and geometric
applications, M. BRION, PREENA SAMUEL, V. UMA
Lectures on the
structure of algebraic groups
and geometric applications
Michel Brion
Institut Fourier, Grenoble, France
Preena Samuel
Chennai Mathematical Institute, Chennai, India.
V. Vma
Indian Institute of Technology Madras, Chennai, India.
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ISBN 978-93-80250-46-5 ISBN 978-93-86279-58-3 (eBook)
DOI 10.1007/978-93-86279-58-3
Preface
The theory of algebraic groups has chiefly been developed along two
distinct directions: linear (or, equivalently, affine) algebraic groups,
and abelian varieties (complete, connected algebraic groups). This
is made possible by a fundamental theorem of Chevalley: any con
nected algebraic group over an algebraically closed field is an ex
tension of an abelian variety by a connected linear algebraic group,
and these are unique.
In these notes, we first expose the above theorem and related
structure results about connected algebraic groups that are neither
affine nor complete. The class of anti-affine algebraic groups (those
having only constant global regular functions) features prominently
in these developments. We then present applications to some ques
tions of algebraic geometry: the classification of complete homoge
neous varieties, and the structure of homogeneous (or translation
invariant) vector bundles and principal bundles over abelian vari
eties.
While the structure theorems presented at the beginning of
these notes go back to the work of Barsotti, Chevalley and Rosen
licht in the 1950's, all the other results are quite recent; they are
mainly due to Sancho de Salas [Sa103, SS09] and the first-named
author [Bri09, BrilOa, Brill, Bri12]. We hope that the present
exposition will stimulate further interest in this domain. In Chap
ter 1, the reader will find a detailed overview of the contents of the
subsequent chapters as well as some open questions.
These notes originate in a series of lectures given at Chennai
Mathematical Institute in January 2011 by the first-named au
thor. He warmly thanks all the attendants of the lectures, espe-
vi
cially V. Balaji, D. S. Nagaraj and C. S. Seshadri, for stimulating
questions and comments; the hospitality of the Institute of Math
ematical Sciences, Chennai, is also gratefully acknowledged. The
three authors wish to thank Balaji for having prompted them to
write up notes of the lectures, and encouraged them along the way;
thanks are also due to T. Szamuely for his very helpful comments
and suggestions on a preliminary version of these notes.
Contents
Preface v
1 Overview 1
1.1 Chevalley's structure theorem 2
1.2 Rosenlicht decomposition ... 4
1.3 Complete homogeneous varieties . 6
1.4 Anti-affine algebraic groups .. 8
1.5 Homogeneous vector bundles .. 10
1.6 Homogeneous principal bundles 12
2 Proof of Chevalley's Theorem 17
2.1 Criteria for affineness ..... 18
2.2 Actions of abelian varieties .. 23
2.3 Rational actions of algebraic groups 27
3 Applications and developments 35
3.1 Some applications . . . . . 35
3.2 Rosenlicht decomposition. . . 41
4 Complete homogeneous varieties 47
4.1 Structure ........... . 48
4.2 Blanchard's lemma for group schemes. 50
4.3 Automorphisms............. 55
5 Anti-affine groups 61
5.1 Characterizations 61
5.2 Structure 64
5.3 Semi-abelian varieties . 65
Vlll CONTENTS
5.4 Extensions by vector groups 68
5.5 Classification ....... . 71
6 Homogeneous vector bundles 75
6.1 Principal bundles . . . . . . 76
6.2 Equivariant automorphisms 80
6.3 Automorphisms of vector bundles 83
6.4 Structure ...... . 87
6.5 Mukai correspondence ..... 91
7 Homogeneous principal bundles 95
7.1 Structure ........... . 96
7.2 Characterizations........ 100
7.3 Special classes of principal bundles 105
Bibliography 111
Index 119
Chapter 1
Overview
Notation and conventions. Throughout these notes, we consider
algebraic varieties and schemes over a fixed algebraically closed base
field k. Unless otherwise stated, schemes are assumed to be of finite
type over k, and points are assumed to be closed, or equivalently
k-rational. For a scheme X, we denote by O(X) the algebra of
global sections of the structure sheaf Ox. A variety is a separated
integral scheme.
A group scheme G is a scheme equipped with morphisms m :
G x G ----> G (the multiplication), i : G ----> G (the inverse) and
with a point ee (the neutral element) which satisfy the axioms of
a group. The neutral component of G is the connected component
containing ee, denoted as GO; this is a normal subgroup scheme of
G, and the quotient G / GO is a finite group scheme.
An algebraic group is a group scheme which is smooth, or equiv
alently, reduced; by a subgroup (scheme) of G, we always mean
a closed subgroup (scheme). Any group scheme G contains a
largest algebraic subgroup, namely, the underlying reduced sub
scheme G
red'
An abelian variety A is a complete connected algebraic group.
It is well-known that such an A is a projective variety, and its
group law is commutative; we denote that law additively, and the
neutral element as OA. For any non-zero illteger n, we denote as
nA the multiplication by n in A, and as An its scheme-theoretic
kernel; recall that nA is an isogeny of A, i.e., a finite surjective
