Homogeneous Spaces and Equivariant Embeddings (eBook)

Acknowledgements 6
Contents 7
Introduction 12
Notation and Conventions 15
1 Algebraic Homogeneous Spaces 19
1 Homogeneous Spaces 19
1.1 Basic Definitions 19
1.2 Tangent Spaces and Automorphisms 21
2 Fibrations, Bundles, and Representations 21
2.1 Homogeneous Bundles 21
2.2 Induction and Restriction 23
2.3 Multiplicities 24
2.4 Regular Representation 24
2.5 Hecke Algebras 25
2.6 Weyl Modules 26
3 Classes of Homogeneous Spaces 28
3.1 Reductions 28
3.2 Projective Homogeneous Spaces 29
3.3 Affine Homogeneous Spaces 29
3.4 Quasiaffine Homogeneous Spaces 31
2 Complexity and Rank 33
4 Local Structure Theorems 33
4.1 Locally Linearizable Actions 33
4.2 Local Structure of an Action 34
4.3 Local Structure Theorem of Knop 37
5 Complexity and Rank of G-varieties 38
5.1 Basic Definitions 38
5.2 Complexity and Rank of Subvarieties 38
5.3 Weight Semigroup 40
5.4 Complexity and Growth of Multiplicities 40
6 Complexity and Modality 42
6.1 Modality of an Action 42
6.2 Complexity and B-modality 43
6.3 Adherence of B-orbits 44
6.4 Complexity and G-modality 45
7 Horospherical Varieties 46
7.1 Horospherical Subgroups and Varieties 46
7.2 Horospherical Type 48
7.3 Horospherical Contraction 48
8 Geometry of Cotangent Bundles 49
8.1 Symplectic Structure 49
8.2 Moment Map 49
8.3 Localization 50
8.4 Logarithmic Version 51
8.5 Image of the Moment Map 51
8.6 Corank and Defect 53
8.7 Cotangent Bundle and Geometry of an Action 54
8.8 Doubled Actions 55
9 Complexity and Rank of Homogeneous Spaces 57
9.1 General Formulæ 57
9.2 Reduction to Representations 59
10 Spaces of Small Rank and Complexity 61
10.1 Spaces of Rank 1 61
10.2 Spaces of Complexity 1 62
11 Double Cones 64
11.1 HV-cones and Double Cones 65
11.2 Complexity and Rank 67
11.3 Factorial Double Cones of Complexity 1 69
11.4 Applications to Representation Theory 70
11.5 Spherical Double Cones 73
3 General Theory of Embeddings 74
12 The Luna--Vust Theory 74
12.1 Equivariant Classification of G-varieties 74
12.2 Universal Model 75
12.3 Germs of Subvarieties 77
12.4 Morphisms, Separation, and Properness 78
13 B-charts 79
13.1 B-charts and Colored Equipment 79
13.2 Colored Data 80
13.3 Local Structure 82
14 Classification of G-models 83
14.1 G-germs 83
14.2 G-models 84
15 Case of Complexity 0 85
15.1 Combinatorial Description of Spherical Varieties 85
15.2 Functoriality 87
15.3 Orbits and Local Geometry 88
16 Case of Complexity 1 89
16.1 Generically Transitive and One-parametric Cases 89
16.2 Hyperspace 90
16.3 Hypercones 92
16.4 Colored Data 94
16.5 Examples 97
16.6 Local Properties 101
17 Divisors 101
17.1 Reduction to B-stable Divisors 101
17.2 Cartier Divisors 102
17.3 Case of Complexity 1 103
17.4 Global Sections of Line Bundles 106
17.5 Ample Divisors 108
18 Intersection Theory 111
18.1 Reduction to B-stable Cycles 111
18.2 Intersection of Divisors 112
18.3 Divisors and Curves 116
18.4 Chow Rings 117
18.5 Halphen Ring 118
18.6 Generalization of the Bézout Theorem 119
4 Invariant Valuations 121
19 G-valuations 122
19.1 Basic Properties 122
19.2 Case of a Reductive Group 123
20 Valuation Cones 124
20.1 Hyperspace 124
20.2 Main Theorem 126
20.3 A Good G-model 126
20.4 Criterion of Geometricity 127
20.5 Proof of the Main Theorem 128
20.6 Parabolic Induction 130
21 Central Valuations 131
21.1 Central Valuation Cone 131
21.2 Central Automorphisms 132
21.3 Valuative Characterization of Horospherical Varieties 134
21.4 G-valuations of a Central Divisor 134
22 Little Weyl Group 135
22.1 Normalized Moment Map 135
22.2 Conormal Bundle to General U-orbits 136
22.3 Little Weyl Group 137
22.4 Relation to Valuation Cones 139
23 Invariant Collective Motion 140
23.1 Polarized Cotangent Bundle 140
23.2 Integration of Invariant Collective Motion 141
23.3 Flats and Their Closures 142
23.4 Non-symplectically Stable Case 144
23.5 Proof of Theorem 22.13 145
23.6 Sources 146
23.7 Root System of a G-variety 147
24 Formal Curves 148
24.1 Valuations via Germs of Curves 148
24.2 Valuations via Formal Curves 149
5 Spherical Varieties 151
25 Various Characterizations of Sphericity 152
25.1 Spherical Spaces 152
25.2 ``Multiplicity-free'' Property 153
25.3 Weakly Symmetric Spaces and Gelfand Pairs 154
25.4 Commutativity 155
25.5 Generalizations 158
26 Symmetric Spaces 161
26.1 Algebraic Symmetric Spaces 161
26.2 -stable Tori 162
26.3 Maximal -fixed Tori 163
26.4 Maximal -split Tori 164
26.5 Classification 166
26.6 Weyl Group 170
26.7 B-orbits 170
26.8 Colored Equipment 171
26.9 Coisotropy Representation 173
26.10 Flats 173
27 Algebraic Monoids and Group Embeddings 174
27.1 Algebraic Monoids 174
27.2 Reductive Monoids 176
27.3 Orbits 178
27.4 Normality and Smoothness 180
27.5 Group Embeddings 181
27.6 Enveloping and Asymptotic Semigroups 184
28 S-varieties 185
28.1 General S-varieties 185
28.2 Affine Case 186
28.3 Smoothness 189
29 Toroidal Embeddings 189
29.1 Toroidal Versus Toric Varieties 190
29.2 Smooth Toroidal Varieties 190
29.3 Cohomology Vanishing 192
29.4 Rigidity 193
29.5 Chow Rings 194
29.6 Closures of Flats 194
30 Wonderful Varieties 195
30.1 Standard Completions 195
30.2 Demazure Embedding 197
30.3 Case of a Symmetric Space 198
30.4 Canonical Class 199
30.5 Cox Ring 199
30.6 Wonderful Varieties 203
30.7 How to Classify Spherical Subgroups 204
30.8 Spherical Spaces of Rank 1 205
30.9 Localization of Wonderful Varieties 207
30.10 Types of Simple Roots and Colors 209
30.11 Combinatorial Classification of Spherical Subgroups and Wonderful Varieties 210
30.12 Proof of the Classification Theorem 212
31 Frobenius Splitting 217
31.1 Basic Properties 217
31.2 Splitting via Differential Forms 218
31.3 Extension to Characteristic Zero 220
31.4 Spherical Case 221
Appendices 223
A Algebraic Geometry 223
A.1 Rational Singularities 223
A.2 Mori Theory 224
A.3 Schematic Points 227
B Geometric Valuations 228
C Rational Modules and Linearization 230
D Invariant Theory 232
E Hilbert Schemes 236
E.1 Classical Case 236
E.2 Nested Hilbert Scheme 238
E.3 Invariant Hilbert Schemes 239
References 242
Name Index 254
Subject Index 258
Notation Index 263