Homogeneous Spaces and Equivariant Embeddings

By: D.A. Timashev

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Hardcover | 7 April 2011

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Homogeneous spaces of linear algebraic groups lie at the crossroads of algebraic geometry, theory of algebraic groups, classical projective and enumerative geometry, harmonic analysis, and representation theory. By standard reasons of algebraic geometry, in order to solve various problems on a homogeneous space, it is natural and helpful to compactify it while keeping track of the group action, i.e., to consider equivariant completions or, more generally, open embeddings of a given homogeneous space. Such equivariant embeddings are the subject of this book. We focus on the classification of equivariant embeddings in terms of certain data of "combinatorial" nature (the Luna-Vust theory) and description of various geometric and representation-theoretic properties of these varieties based on these data. The class of spherical varieties, intensively studied during the last three decades, is of special interest in the scope of this book. Spherical varieties include many classical examples, such as Grassmannians, flag varieties, and varieties of quadrics, as well as well-known toric varieties. We have attempted to cover most of the important issues, including the recent substantial progress obtained in and around the theory of spherical varieties.

Industry Reviews

From the reviews:

"The book by D. A. Timashev is a welcome survey of old and new results on actions of algebraic groups on algebraic varieties. ... The book contains an extensive list of references, both to original research articles as well as to books and monographs. It is surely a welcome guide for nonexperts who want to enter the subject and a good reference for the specialist." (Gestur "lafsson, Bulletin of the American Mathematical Society, Vol. 51 (2), April, 2014)

"This book is a survey of the study of equivariant embeddings of homogeneous spaces for connected reductive algebraic groups. ... The book is well written and puts together many results in a complete and concise manner." (Lucy Moser-Jauslin, Mathematical Reviews, Issue 2012 e)

"The book under review is concerned with the study of equivariant embeddings of homogeneous spaces ... . This monograph provides a compact survey of these issues, including many results of Brion, Knop, Luna, and Vust, among others, as well as the author's own significant contributions to the field. ... The author admirably fills a gap in the current literature on algebraic groups with this book. It should make a welcome addition to any mathematician's library." (Nathan Ilten, Zentralblatt MATH, Vol. 1237, 2012)

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Introduction	p. xv
Notation and Conventions	p. xviii
Algebraic Homogeneous Spaces	p. 1
Homogeneous Spaces	p. 1
Basic Definitions	p. 1
Tangent Spaces and Automorphisms	p. 3
Fibrations, Bundles, and Representations	p. 3
Homogeneous Bundles	p. 3
Induction and Restriction	p. 5
Multiplicities	p. 6
Regular Representation	p. 6
Hecke Algebras	p. 7
Weyl Modules	p. 8
Classes of Homogeneous Spaces	p. 10
Reductions	p. 10
Projective Homogeneous Spaces	p. 11
Affine Homogeneous Spaces	p. 11
Quasiaffine Homogeneous Spaces	p. 13
Complexity and Rank	p. 15
Local Structure Theorems	p. 15
Locally Linearizable Actions	p. 15
Local Structure of an Action	p. 16
Local Structure Theorem of Knop	p. 19
Complexity and Rank of G-varieties	p. 20
Basic Definitions	p. 20
Complexity and Rank of Subvarieties	p. 20
Weight Semigroup	p. 22
Complexity and Growth of Multiplicities	p. 22
Complexity and Modality	p. 24
Modality of an Action	p. 24
Complexity and B-modality	p. 25
Adherence of B-orbits	p. 26
Complexity and G-modality	p. 27
Horospherical Varieties	p. 28
Horospherical Subgroups and Varieties	p. 28
Horospherical Type	p. 30
Horospherical Contraction	p. 30
Geometry of Cotangent Bundles	p. 31
Symplectic Structure	p. 31
Moment Map	p. 31
Localization	p. 32
Logarithmic Version	p. 33
Image of the Moment Map	p. 33
Corank and Defect	p. 35
Cotangent Bundle and Geometry of an Action	p. 36
Doubled Actions	p. 37
Complexity and Rank of Homogeneous Spaces	p. 39
General Formulae	p. 39
Reduction to Representations	p. 41
Spaces of Small Rank and Complexity	p. 43
Spaces of Rank ≤ 1	p. 43
Spaces of Complexity ≤ 1	p. 44
Double Cones	p. 46
HV-cones and Double Cones	p. 47
Complexity and Rank	p. 49
Factorial Double Cones of Complexity ≤ 1	p. 51
Applications to Representation Theory	p. 52
Spherical Double Cones	p. 55
General Theory of Embeddings	p. 57
The Luna-Vust Theory	p. 57
Equivariant Classification of G-varieties	p. 57
Universal Model	p. 58
Germs of Subvarieties	p. 60
Morphisms, Separation, and Properness	p. 61
B-charts	p. 62
B-charts and Colored Equipment	p. 62
Colored Data	p. 63
Local Structure	p. 65
Classification of G-models	p. 66
G-germs	p. 66
G-models	p. 67
Case of Complexity 0	p. 68
Combinatorial Description of Spherical Varieties	p. 68
Functoriality	p. 70
Orbits and Local Geometry	p. 71
Case of Complexity 1	p. 72
Generically Transitive and One-parametric Cases	p. 72
Hyperspace	p. 73
Hypercones	p. 75
Colored Data	p. 77
Examples	p. 80
Local Properties	p. 84
Divisors	p. 84
Reduction to B-stable Divisors	p. 84
Cartier Divisors	p. 85
Case of Complexity ≤ 1	p. 86
Global Sections of Line Bundles	p. 89
Ample Divisors	p. 91
Intersection Theory	p. 94
Reduction to B-stable Cycles	p. 94
Intersection of Divisors	p. 95
Divisors and Curves	p. 99
Chow Rings	p. 100
Halphen Ring	p. 101
Generalization of the Bézout Theorem	p. 102
Invariant Valuations	p. 105
G-valuations	p. 106
Basic Properties	p. 106
Case of a Reductive Group	p. 107
Valuation Cones	p. 108
Hyperspace	p. 108
Main Theorem	p. 110
A Good G-model	p. 110
Criterion of Geometricity	p. 111
Proof of the Main Theorem	p. 112
Parabolic Induction	p. 114
Central Valuations	p. 115
Central Valuation Cone	p. 115
Central Automorphisms	p. 116
Valuative Characterization of Horospherical Varieties	p. 118
G-valuations of a Central Divisor	p. 118
Little Weyl Group	p. 119
Normalized Moment Map	p. 119
Conormal Bundle to General U-orbits	p. 120
Little Weyl Group	p. 121
Relation to Valuation Cones	p. 123
Invariant Collective Motion	p. 124
Polarized Cotangent Bundle	p. 124
Integration of Invariant Collective Motion	p. 125
Flats and Their Closures	p. 126
Non-symplectically Stable Case	p. 128
Proof of Theorem 22.13	p. 129
Sources	p. 130
Root System of a G-variety	p. 131
Formal Curves	p. 132
Valuations via Germs of Curves	p. 132
Valuations via Formal Curves	p. 133
Spherical Varieties	p. 135
Various Characterizations of Sphericity	p. 136
Spherical Spaces	p. 136
ôMultiplicity-freeö Property	p. 137
Weakly Symmetric Spaces and Gelfand Pairs	p. 138
Commutativity	p. 139
Generalizations	p. 142
Symmetric Spaces	p. 145
Algebraic Symmetric Spaces	p. 145
-stable Tori	p. 146
Maximal -fixed Tori	p. 147
Maximal -split Tori	p. 148
Classification	p. 150
Weyl Group	p. 154
B-orbits	p. 154
Colored Equipment	p. 155
Coisotropy Representation	p. 157
Flats	p. 157
Algebraic Monoids and Group Embeddings	p. 158
Algebraic Monoids	p. 158
Reductive Monoids	p. 160
Orbits	p. 162
Normality and Smoothness	p. 164
Group Embeddings	p. 165
Enveloping and Asymptotic Semigroups	p. 168
S-varieties	p. 169
General S-varieties	p. 169
Affine Case	p. 170
Smoothness	p. 173
Toroidal Embeddings	p. 173
Toroidal Versus Toric Varieties	p. 174
Smooth Toroidal Varieties	p. 174
Cohomology Vanishing	p. 176
Rigidity	p. 177
Chow Rings	p. 178
Closures of Flats	p. 178
Wonderful Varieties	p. 179
Standard Completions	p. 179
Demazure Embedding	p. 181
Case of a Symmetric Space	p. 182
Canonical Class	p. 183
Cox Ring	p. 183
Wonderful Varieties	p. 187
How to Classify Spherical Subgroups	p. 188
Spherical Spaces of Rank 1	p. 189
Localization of Wonderful Varieties	p. 191
Types of Simple Roots and Colors	p. 193
Combinatorial Classification of Spherical Subgroups and Wonderful Varieties	p. 194
Proof of the Classification Theorem	p. 196
Frobenius Splitting	p. 201
Basic Properties	p. 201
Splitting via Differential Forms	p. 202
Extension to Characteristic Zero	p. 204
Spherical Case	p. 205
Appendices	p. 207
Algebraic Geometry	p. 207
Rational Singularities	p. 207
Mori Theory	p. 208
Schematic Points	p. 211
Geometric Valuations	p. 212
Rational Modules and Linearization	p. 214
Invariant Theory	p. 216
Hilbert Schemes	p. 220
Classical Case	p. 220
Nested Hilbert Scheme	p. 222
Invariant Hilbert Schemes	p. 223
References	p. 227
Name Index	p. 239
Subject Index	p. 243
Notation Index	p. 249
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Homogeneous Spaces and Equivariant Embeddings

At a Glance

Hardcover

Industry Reviews

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