Topological Methods in Algebraic Transformation Groups: Proceedings of a Conference at Rutgers University

215 Pages·1989·6.605 MB·English

by  Hanspeter Kraft, Ted Petrie (auth.), Hanspeter Kraft, Ted Petrie, Gerald W. Schwarz (eds.)

Progress in Mathematics
Volume 80
Series Editors
1. Oesterle
A. Weinstein
Topological Methods
in Algebraic
Transformation Groups
Proceedings of a Conference at
Rutgers University
Edited by
Hanspeter Kraft
Ted Petrie
Gerald W. Schwarz
1989 Birkhauser
Boston . Basel . Berlin
Hanspeter Kraft Ted Petrie
Mathematisches Institut Department of Mathematics
Universitat Basel Rutgers University
Rheinsprung 21 New Brunswick, New Jersey 07102
CH-4051 Basel U.SA
Switzerland
Gerald W. Schwarz
Department of Mathematics
Brandeis University
Waltham, Massachusetts 02254-9110
U.SA.
ISBN-13: 978-1-4612-8219-8 e-ISBN-13: 978-1-4612-3702-0
DOl: 10.1007/978-1-4612-3702-0
Library of Congress Cataloging-in-Publication Data
Topological methods in algebraic transformation group / [edited by]
Hanspeter Kraft, Ted Petrie, Gerald W. Schwarz.
p. cm. - (progress in mathematics; v. SO)
Papers from the conference, "Topological Methods in Algebraic
Transformation Groups" held at Rutgers University, 4-8 April, 1988.
1. Transformation groups-Congresses. 2. Algebraic topology
-Congresses. 3. Geometry, Algebraic-Congresses. I. Kraft,
Hanspeter, 1944- . II. Petrie, Ted, 1939- . III. Schwarz,
Gerald W., 1946- . IV. Series: Progress in mathematics (Boston, Mass.) ;
vol. SO
QA385.T67 1989
514'.2-dc20
Printed on acid-free paper.
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9 8 765 4 3 2 1
PREFACE
In recent years, there has been increasing interest and activity in the
area of group actions on affine and projective algebraic varieties. Tech
niques from various branches of mathematics have been important for this
study, especially those coming from the well-developed theory of smooth
compact transformation groups. It was timely to have an interdisciplinary
meeting on these topics.
We organized the conference "Topological Methods in Alg~braic
Transformation Groups," which was held at Rutgers University, 4-8 April,
1988. Our aim was to facilitate an exchange of ideas and techniques among
mathematicians studying compact smooth transformation groups, alge
braic transformation groups and related issues in algebraic and analytic
geometry. The meeting was well attended, and these Proceedings offer
a larger audience the opportunity to benefit from the excellent survey
and specialized talks presented. The main topics concerned various as
pects of group actions, algebraic quotients, homogeneous spaces and their
compactifications.
The meeting was made possible by support from Rutgers University
and the National Science Foundation. We express our deep appreciation
for this support. We also thank Annette Neuen for her assistance with
the technical preparation of these Proceedings.
The Editors.
TABLE OF CONTENTS
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Linearizing flat families of reductive group representations. . . . . . . . . . 5
HYMAN BASS
Spherical varieties: An introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
MICHEL BRION
Homology planes: An announcement and survey. . . . . . . . . . . . . . . . . . .. 27
TAMMO TOM DIECK, TED PETRIE
Fixed point free algebraic actions on varieties diffeomorphic to R n .• 49
HEINER DOVERMANN, MIKIYA MASUDA, TED PETRIE
Algebraic automorphisms of affine space. ....... ............... ..... 81
HANSPETER KRAFT
Almost homogeneous Artin-Moisezon varieties under the action
of PSL (C) ................................................... 107
2
DOMINGO LUNA, LUCY MOSER-JAUSLIN, THIERRY VUST
On the topology of curves in complex surfaces. . . . . . . . . . . . . . . . . . . . .. 11 7
WALTER D. NEUMANN
The topology of algebraic quotients ................................ 135
GERALD W. SCHWARZ
Rationality of moduli spaces via invariant theory ................... 153
NICHOLAS I. SHEPHERD-BARRON
Unipotent actions on affine space .................................. 165
DENNIS M. SNOW
Algebraic characterization of the affine plane and the affine
3-space ........................................................ 177
TORU SUGIE
Classification of 3-dimensional homogeneous complex manifolds ..... 191
JORG WINKELMANN
INTRODUCTION
The subject of smooth transformation groups has been strongly in
fluenced by the following two central problems:
• Smooth Linearization Problem: Is every smooth action of a
compact Lie group on Euclidean space conjugate to a linear action?
• Smooth Fixed Point Problem: Does every smooth action of a
compact Lie group on Euclidean space have a fixed point?
The tools used to settle these problems-and the further problems they
generated-have been important themes in the field. During the last few
years researchers from algebraic transformation groups, as well as those
from smooth transformation groups, have recognized the interest and im
portance of the analogues of these questions in the algebraic category:
• Algebraic Linearization Problem: Is every algebraic action of
a reductive group on affine space conjugate to a linear action?
• Algebraic Fixed Point Problem: Does every algebraic action of
a reductive group on affine space have a fixed point?
The intention of the Rutgers Conference on "Topological Methods in Al
gebraic Transformation Groups" was to facilitate an exchange of ideas
among mathematicians whose expertise includes smooth, analytic or al
gebraic transformation groups. The focus was on the two problems above.
Experience from smooth transformation groups shows that their study re
quires a wide range of techniques and topics:
• Group actions on affine space.
• Characterization and structure of affine space.
• Prehomogeneous varieties.
• Orbit spaces and quotients.
• G-vector bundles and K-theory.
Below we discuss the connection of the topics to the conference theme
and to the papers in this volume.
Since the time of the conference SCHWARZ has constructed the first
non-linearizable actions of reductive groups on affine spaces, giving a neg
ative answer to the Algebraic Linearization Problem. The Algebraic Fixed
Point Problem is still open.
2 Introduction
Group actions on affine space. The survey of KRAFT provides a broad
introduction to the current problems in the theory of algebraic transfor
mation groups. Emphasis is on the case of reductive group actions on afine
space. It is a good place to get an overview of the subject.
SNOW's survey is about actions of unipotent groups. These groups
are, in some sense, those furthest removed from being reductive. An im
cn.
portant question concerns the structure of free C-actions on Is every
such action isomorphic to translation by a fixed vector? SNOW reviews
both positive results and recent counterexamples.
The paper of DOVERMANN-MASUDA-PETRIE shows how to construct
fixed point free real algebraic actions of the icosahedral group on real
varieties diffeomorphic to R n. These are algebraic versions of previous
(purely COO) examples.
Characterization and structure of affine space. We begin by men
tioning the following simple characterization of Euclidean space of di
mension at least 5 in the smooth category. It is essential in treating the
Smooth Linearization Problem.
Theorem. For n at least 5, real Euclidean n-space is the only smooth
contractible manifold which is simply connected at infinity.
This theorem can be used to construct a fixed point free smooth action
of the icosahedral group on Euclidean space (see BREDON, "Introduction
to Compact Transformation Groups").
c
In the algebraic setting, the analogous problem is to characterize n
among algebraic varieties. SUGlE's paper is concerned with the cases n = 2
=
and n 3. He reviews topological and algebraic results and techniques. He
discusses applications and related problems (e.g., Zariski's Cancellation
Problem).
The paper of TOMDIECK-PETRIE deals with homology planes, i.e.,
non-singular, acyclic, affine surfaces over C. They show how to construct
all such surfaces starting from certain configurations of curves in the pro
jective plane. They make progress towards proving the following conjec
ture of PETRIE:
Conjecture. The only homology plane with a non-trivial algebraic auto
morphism group is C2•
A positive answer to this conjecture would imply that every C* -action on
C3 is linearizable (see the survey of KRAFT).
Introd uction 3
NEUMANN studies the topological classification of curves which are
embedded in complex surfaces, and, in particular, curves embedded in
C2• He determines the list of isomorphism types for embedded curves of
low genus.
Pre homogeneous varieties. The articles of WINKELMANN, BRION and
LUNA-MoSER-VUST concern the structure of homogeneous and preho
mogeneous varieties: A G-variety is called prehomogeneous if it contains
a dense orbit. Prehomogeneous varieties are those whose structure one
should try to understand first. BRION gives a survey and introduction to
spherical varieties: A normal G-variety X, where G is a reductive algebraic
group, is said to be spherical if a Borel subgroup of G has a dense orbit.
Spherical varieties include compactifications of symmetric spaces. BRION
points out many of the beautiful properties and uses of these varieties.
The paper of LUNA-MoSER-VUST concerns the structure of spaces
which are prehomogeneous with respect to an action of PSL (C). It
2
shows that-somewhat surprisingly-there are examples which are Artin
Moisezon but not algebraic. Artin-Moisezon spaces already arise as quo
tients of non-affine varieties by finite group actions.
WINKELMANN reviews his classification of all homogeneous complex
manifolds of dimension 3 or less. There are examples which are homo
geneous with respect to a real Lie group, but not with respect to any
complex Lie group.
Orbit spaces and quotients. In order to study the action of an algebraic
group on a variety, it is important to be able to form a quotient and
study its properties. The survey article of SCHWARZ reviews many of the
important topics in the theory of algebraic quotients of affine G-varieties.
He reproves the basic results of LUNA, NEEMAN and KRAFT-PETRIE
RANDALL on the algebraic, analytic and topological structure of quotients.
SHEPHERD-BARRON's survey treats the following question: When is
the quotient variety rational? That is, when is its function field a purely
transcendental extension of C? SHEPHERD-BARRON gives examples of
various techniques and their application to classical moduli problems.
G-vector bundles and K-theory. There are several important results
known about the theory of algebraic G-vector bundles over G-varieties.
We concentrate on the case that the base is a representation of G. Results
of BASS-HABOUSH show that all such G-vector bundles are stably trivial,
and results of KRAFT show that all such bundles are equivariantly locally
trivial (in the Zariski topology). Surprisingly, the recent counterexamples
4 Introduction
of SCHWARZ to the Algebraic Linearization Problem arise from non-trivial
G-vector bundles over representations (see KRAFT's survey).
BASS's paper concerns the problem of characterizing G-vector bun
dles among G-varieties X over a given variety S. IT X is flat over S and
every fiber is isomorphic to a representation of G, then one would like to
conclude that X is isomorphic to a G-vector bundle over S. BASS shows
that this is "stably" true, i.e., X x V is a G-vector bundle over S for some
G-representatioh V.