Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action

247 Pages·2002·12.69 MB·English

by  Andrzej Białynicki-Birula

Encyclopaedia of Mathematical Sciences
Volume 131
Invariant Theory and Algebraic Transformationgroups
Subseries Editors:
R. V. Gamkrelidze V.L. Popov
Springer-Verlag Berlin Heidelberg GmbH
A. Bialynicki-Birula J. B. Carrell W. M. McGovern
Algebraic Quotients.
Torus Actions
and Cohomology.
The Adjoint Representation
and the Adjoint Action
Springer
Andrzej Bialynicki-Birula
Warsaw University
Department of Mathematics and Computer Science
Banacha2
02-097 Warsaw
Poland
e-mail: [email protected]
James B. Carrell
The University of British Columbia
Department of Mathematics
1984 Mathematics Road
Vancouver V 6T 1Z2
Canada
e-mail: [email protected]
William M. McGovern
U niversity of Washington
Department of Mathematics
P.O. Box 354350
Seattle, W A 98195
USA
e-mail: [email protected]
Founding Editor of the Encyclopaedia of Mathematical Sciences:
R.V. Gamkrelidze
Mathematics Subject Classification (2000):
14L30, 14LIO, 14L15, 14L17, 14L24, 14F25, 14F43, 14022, 13A50, 14M17, 14M25, 14M20, 22E46, 22E60, 22E20, 22E25, 20G05,
20G15, 22E45, 22E46
ISSN 0938-0396
ISBN 978-3-642-07745-6 ISBN 978-3-662-05071-2 (eBook)
DOI 10.1007/978-3-662-05071-2
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Contents
I. Quotients by Actions of Groups
Andrzej Bialynicki-Birula
1
ll. Torus Actions and Cohomology
James B. Carrell
83
llI. The Adjoint Representation and the Adjoint Action
William M. McGovern
159
SUbject Index
239
I. Quotients by Actions of Groups
Andrzej Bialynicki-Birula
Contents
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§2.1. Spaces, Schemes, Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§2.2. Preequivalence Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§2.3. Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
§2.4. Group Actions ............................................. 8
Chapter 3 Basic Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Chapter 4 Standard Examples and Applications. . . . . . . . . . . . . . . . . . . . . . . . 12
§4.1. Principal Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
§4.2. Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
§4.3. Quotients by Finite Group Actions........... ... . . ... .......... 13
§4.4. Coset Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
§4.5. Quotients of Affine Spaces by Linear Actions of Groups. . . . . . . . . . 16
§4.6. Induced Actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter 5 The Affine Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Chapter 6 Mumford's G.I.T. ........................................ 18
§6.1. General Case............................................... 19
§6.2. Case of a Projective Space ................................... 20
A. Białynicki-Birula et al., Algebraic Quotients. Torus Actions and Cohomology.
The Adjoint Representation and the Adjoint Action
© Springer-Verlag Berlin Heidelberg 2002
2 Andrzej Bialynicki-Birula
§6.3. Case of a Projective Variety .................................. 21
§6.4. Hilbert-Mumford Criterium.................................. 21
§6.5. The Case of Actions of Any Linear Algebraic Group ............. 22
Chapter 7 Good Quotients . . . . . . . . . . . . . . . . . . . . . .. .. . .. . .. . .. .. . .. ... 23
§7.1. General Properties. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
§7.2. G-Maximal Subsets ........................................ 25
§7.3. Variation of Stability........................................ 27
Chapter 8 Proper Actions and Quotients by Proper Actions .............. 28
§8.1. Quotients by Closed Equivalence Relations ..................... 28
§8.2. Existence of Quotients by Proper Group Actions. . . . . . . . . . . . . . . .. 29
§8.3. Existence of Quotients of Proper Groupoids . . . . . . . . . . . . . . . . . . . . . 30
§8.4. Seshadri Cover............................................. 31
Chapter 9 Stacks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Chapter 10 Complex Analytic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
§ 10. 1. Quotients by Proper Actions and Equivalence Relations. . . . . . . . . . . 37
§10.2.Stein and Good Quotients.................................... 38
§10.3.MomentMaps.............................................. 39
Chapter 11 Good Quotients by Actions of Tori. . . . . . .. . . . . . .. . . . . . .. . . .. 42
§11.1.Quotients by Actions of One-Dimensional Tori.................. 42
§ 11.2. Quotients by Actions of Tori of Arbitrary Dimension . . . . . . . . . . . .. 47
§ 11.3. Quotients of Open Subsets of Projective and Affine Spaces . . . . . . . . 51
§ 11.4. Good Quotients of Toric Varieties by Actions of Tori .... . . . . . . . . . 53
Chapter 12 Hilbert-Mumford Type Theorems.......................... 54
Chapter 13 Chow and Hilbert Quotients ............................... 56
Chapter 14 Categorical Quotients..................................... 57
§14.1.Categorical Quotients....................................... 57
§ 14.2. Quotient Morphisms ........................................ 59
Chapter 15 Sections, Slices and Reductions ............................ 59
Chapter 16 Local and Global Properties of Quotients. . . . . . . . . . . . . . . . . . . .. 64
I. Quotients by Actions of Groups 3
§ 16. 1. Local Properties of Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64
§16.2.Global Properties of Quotients ......... ............ . .......... 65
Chapter 17 Applications to the Theory of Moduli. . . . . . . . . . . . . . . . . . . . . . .. 69
Chapter 18 Final Remarks........................................... 76
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
4 Andrzej Bialynicki-Birula
Chapter 1
Introduction
The aim of this survey is to present the main trends and directions of research
in the theory of quotients by group actions. The theory, in the assumed here sense,
contains Geometric Invariant Theory understood as a theory providing geometric
interpretations of rings and more generally sheaves of invariant functions determined
by group actions. Though the affine (i.e. local) case plays the basic role in the theory,
we shall put attention mainly on results of global character. The affine case should
be considered rather as a part of Classical Invariant Theory, because translation of
algebraic theory of rings of invariants into the geometric language is usually a matter
of routine.
We shall mainly consider the case of actions of algebraic groups on algebraic
varieties over an algebraically closed base field of characteristic O. As quotient spaces
we shall accept algebraic spaces over the field. But, though the results presented
here and constituting a skeleton of the theory will be presented under these unified
assumptions, the definitions will be formulated in the general case (of schemes or
algebraic spaces) and, when in some cases theorems concerning this general case
are known, the generalizations will be mentioned. It seems that this way is justified
because, on one hand, various authors did not insist on presenting their results in
the most general possible form, but stated assumptions of the theorems according to
their taste or circumstances. On the other hand, changes of assumptions in various
fragments of the theory would probably make difficult for the reader to trace the
development of the theory as a whole. In some chapters we shall admit common
assumptions for all stated there results, but, usually, for convenience of the reader,
we shall recall the assumptions in formulated there theorems.
We shall use freely needed notions and results from outside the theory of quotients.
In chapter 2 we fix notation and gather definitions which are the most fundamental
for presenting the theory.
The theory of quotients by actions of groups has in Algebraic Geometry important
and interesting applications in studying moduli problems in many various special
cases. In fact, Mumford creating his G.I.T. had this applications in mind. However,
later the theory has developed as a part of the theory of algebraic groups and their
actions and has its own aims and problems independent on possible applications to
constructing and describing moduli spaces. In the survey we shall follow this second
point of view. Examples of moduli spaces will be mentioned in the text very rarely.
However, in the second to the last chapter we shall describe some simple and the
most typical applications of the theory of quotients to various moduli problems.
Terminology used here, though maybe, not consistent, has already some historical
background and tradition. The terminology of the analytic case is adapted from
Algebraic Geometry and does not agree with one introduced in some papers on the
subject.
I. Quotients by Actions of Groups 5
Bibliography mentioned in the text and then gathered at the end of the survey is
not complete. However the reader will have no troubles in tracing all pertinent papers
consulting the bibliography provided in the papers with a reference given here in
bold face.
The author wants to express his thanks to Polish Commitee of Research (KBN)
and Polish Foundation of Sciences (FNP), for partial support of the project.
Chapter 2
Terminology and Notation
§2.1. Spaces, Schemes, Varieties
We fix a locally noetherian base scheme S. All schemes (more generally: algebraic
spaces) will be S-schemes (algebraic S-spaces) of finite type. All morphisms (products
etc) are supposed to be S-morphisms (S-products, etc).
By k we denote a fixed algebraically closed field. By an algebraic k-variety we
mean a reduced Spec(k)-scheme (of finite type over Spec(k». Next, pn(k), (An(k),
vn (k» denote the n-dimensional projective (affine, vector, respectively) space over k.
When there is no danger of ambiguity we shall sayan algebraic variety (space)
instead of an algebraic k-variety (k-space, respectively). Similarly we shall write
pn, vn instead of pn(k), vn(k), respectively.
For brevity, instead of "an algebraic S-space" we shall usually say "an S-space"
and we shall reserve the name "algebraic space" for S-spaces, where S = Spec(k).
Adding prefix "pre" to "space", "scheme" or "variety" means that the assumption
that the object is separated has been dropped.
Let X and W be two S-schemes. Then by X (W) we denote the set of all W -rational
points of X, i.e. the set of all S-morphisms W -+ X. In case, S = Spec(k), W =
Spec (P), where P is a k-algebra, we shall write X(P) in place of X(Spec (P». Any
S-morphism ¢ : WI -+ W2 induces a map X(W2) -+ X(WI) denoted by X(¢). In
this way X defines a (contravariant) functor from the category of S-schemes into the
category of sets. The functor determines X and is denoted by X. If ¢ : X I -+ X 2 is
an S-morphism, then the induced map Xl (W) -+ X2(W) is denoted by ¢(W). The
family {¢(W) }q" where ¢ runs over the set of all morphisms X I -+ X2, determines
a natural transformation of functors Xl=> X2.
§2.2. Preequivalence Relations
e
Let be a fixed subcategory of the category of S-spaces (of finite type) and their
S-morphisms. The most interesting and important examples are the whole category