Timashev Homogeneous Spaces and Equivariant Embeddings

2011
ISBN: 978-3-642-18399-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 138, 254 Seiten

Reihe: Encyclopaedia of Mathematical Sciences

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.

Autoren/Hrsg.

Timashev, D. A.

Inhaltsverzeichnis

1;Acknowledgements;6
2;Contents;7
3;Introduction;12
3.1;Notation and Conventions;15
4;1 Algebraic Homogeneous Spaces;19
4.1;1 Homogeneous Spaces;19
4.1.1;1.1 Basic Definitions;19
4.1.2;1.2 Tangent Spaces and Automorphisms;21
4.2;2 Fibrations, Bundles, and Representations;21
4.2.1;2.1 Homogeneous Bundles;21
4.2.2;2.2 Induction and Restriction;23
4.2.3;2.3 Multiplicities;24
4.2.4;2.4 Regular Representation;24
4.2.5;2.5 Hecke Algebras;25
4.2.6;2.6 Weyl Modules;26
4.3;3 Classes of Homogeneous Spaces;28
4.3.1;3.1 Reductions;28
4.3.2;3.2 Projective Homogeneous Spaces;29
4.3.3;3.3 Affine Homogeneous Spaces;29
4.3.4;3.4 Quasiaffine Homogeneous Spaces;31
5;2 Complexity and Rank;33
5.1;4 Local Structure Theorems;33
5.1.1;4.1 Locally Linearizable Actions;33
5.1.2;4.2 Local Structure of an Action;34
5.1.3;4.3 Local Structure Theorem of Knop;37
5.2;5 Complexity and Rank of G-varieties;38
5.2.1;5.1 Basic Definitions;38
5.2.2;5.2 Complexity and Rank of Subvarieties;38
5.2.3;5.3 Weight Semigroup;40
5.2.4;5.4 Complexity and Growth of Multiplicities;40
5.3;6 Complexity and Modality;42
5.3.1;6.1 Modality of an Action;42
5.3.2;6.2 Complexity and B-modality;43
5.3.3;6.3 Adherence of B-orbits;44
5.3.4;6.4 Complexity and G-modality;45
5.4;7 Horospherical Varieties;46
5.4.1;7.1 Horospherical Subgroups and Varieties;46
5.4.2;7.2 Horospherical Type;48
5.4.3;7.3 Horospherical Contraction;48
5.5;8 Geometry of Cotangent Bundles;49
5.5.1;8.1 Symplectic Structure;49
5.5.2;8.2 Moment Map;49
5.5.3;8.3 Localization;50
5.5.4;8.4 Logarithmic Version;51
5.5.5;8.5 Image of the Moment Map;51
5.5.6;8.6 Corank and Defect;53
5.5.7;8.7 Cotangent Bundle and Geometry of an Action;54
5.5.8;8.8 Doubled Actions;55
5.6;9 Complexity and Rank of Homogeneous Spaces;57
5.6.1;9.1 General Formulæ;57
5.6.2;9.2 Reduction to Representations;59
5.7;10 Spaces of Small Rank and Complexity;61
5.7.1;10.1 Spaces of Rank 1;61
5.7.2;10.2 Spaces of Complexity 1;62
5.8;11 Double Cones;64
5.8.1;11.1 HV-cones and Double Cones;65
5.8.2;11.2 Complexity and Rank;67
5.8.3;11.3 Factorial Double Cones of Complexity 1;69
5.8.4;11.4 Applications to Representation Theory;70
5.8.5;11.5 Spherical Double Cones;73
6;3 General Theory of Embeddings;74
6.1;12 The Luna--Vust Theory;74
6.1.1;12.1 Equivariant Classification of G-varieties;74
6.1.2;12.2 Universal Model;75
6.1.3;12.3 Germs of Subvarieties;77
6.1.4;12.4 Morphisms, Separation, and Properness;78
6.2;13 B-charts;79
6.2.1;13.1 B-charts and Colored Equipment;79
6.2.2;13.2 Colored Data;80
6.2.3;13.3 Local Structure;82
6.3;14 Classification of G-models;83
6.3.1;14.1 G-germs;83
6.3.2;14.2 G-models;84
6.4;15 Case of Complexity 0;85
6.4.1;15.1 Combinatorial Description of Spherical Varieties;85
6.4.2;15.2 Functoriality;87
6.4.3;15.3 Orbits and Local Geometry;88
6.5;16 Case of Complexity 1;89
6.5.1;16.1 Generically Transitive and One-parametric Cases;89
6.5.2;16.2 Hyperspace;90
6.5.3;16.3 Hypercones;92
6.5.4;16.4 Colored Data;94
6.5.5;16.5 Examples;97
6.5.6;16.6 Local Properties;101
6.6;17 Divisors;101
6.6.1;17.1 Reduction to B-stable Divisors;101
6.6.2;17.2 Cartier Divisors;102
6.6.3;17.3 Case of Complexity 1;103
6.6.4;17.4 Global Sections of Line Bundles;106
6.6.5;17.5 Ample Divisors;108
6.7;18 Intersection Theory;111
6.7.1;18.1 Reduction to B-stable Cycles;111
6.7.2;18.2 Intersection of Divisors;112
6.7.3;18.3 Divisors and Curves;116
6.7.4;18.4 Chow Rings;117
6.7.5;18.5 Halphen Ring;118
6.7.6;18.6 Generalization of the Bézout Theorem;119
7;4 Invariant Valuations;121
7.1;19 G-valuations;122
7.1.1;19.1 Basic Properties;122
7.1.2;19.2 Case of a Reductive Group;123
7.2;20 Valuation Cones;124
7.2.1;20.1 Hyperspace;124
7.2.2;20.2 Main Theorem;126
7.2.3;20.3 A Good G-model;126
7.2.4;20.4 Criterion of Geometricity;127
7.2.5;20.5 Proof of the Main Theorem;128
7.2.6;20.6 Parabolic Induction;130
7.3;21 Central Valuations;131
7.3.1;21.1 Central Valuation Cone;131
7.3.2;21.2 Central Automorphisms;132
7.3.3;21.3 Valuative Characterization of Horospherical Varieties;134
7.3.4;21.4 G-valuations of a Central Divisor;134
7.4;22 Little Weyl Group;135
7.4.1;22.1 Normalized Moment Map;135
7.4.2;22.2 Conormal Bundle to General U-orbits;136
7.4.3;22.3 Little Weyl Group;137
7.4.4;22.4 Relation to Valuation Cones;139
7.5;23 Invariant Collective Motion;140
7.5.1;23.1 Polarized Cotangent Bundle;140
7.5.2;23.2 Integration of Invariant Collective Motion;141
7.5.3;23.3 Flats and Their Closures;142
7.5.4;23.4 Non-symplectically Stable Case;144
7.5.5;23.5 Proof of Theorem 22.13;145
7.5.6;23.6 Sources;146
7.5.7;23.7 Root System of a G-variety;147
7.6;24 Formal Curves;148
7.6.1;24.1 Valuations via Germs of Curves;148
7.6.2;24.2 Valuations via Formal Curves;149
8;5 Spherical Varieties;151
8.1;25 Various Characterizations of Sphericity;152
8.1.1;25.1 Spherical Spaces;152
8.1.2;25.2 ``Multiplicity-free'' Property;153
8.1.3;25.3 Weakly Symmetric Spaces and Gelfand Pairs;154
8.1.4;25.4 Commutativity;155
8.1.5;25.5 Generalizations;158
8.2;26 Symmetric Spaces;161
8.2.1;26.1 Algebraic Symmetric Spaces;161
8.2.2;26.2 -stable Tori;162
8.2.3;26.3 Maximal -fixed Tori;163
8.2.4;26.4 Maximal -split Tori;164
8.2.5;26.5 Classification;166
8.2.6;26.6 Weyl Group;170
8.2.7;26.7 B-orbits;170
8.2.8;26.8 Colored Equipment;171
8.2.9;26.9 Coisotropy Representation;173
8.2.10;26.10 Flats;173
8.3;27 Algebraic Monoids and Group Embeddings;174
8.3.1;27.1 Algebraic Monoids;174
8.3.2;27.2 Reductive Monoids;176
8.3.3;27.3 Orbits;178
8.3.4;27.4 Normality and Smoothness;180
8.3.5;27.5 Group Embeddings;181
8.3.6;27.6 Enveloping and Asymptotic Semigroups;184
8.4;28 S-varieties;185
8.4.1;28.1 General S-varieties;185
8.4.2;28.2 Affine Case;186
8.4.3;28.3 Smoothness;189
8.5;29 Toroidal Embeddings;189
8.5.1;29.1 Toroidal Versus Toric Varieties;190
8.5.2;29.2 Smooth Toroidal Varieties;190
8.5.3;29.3 Cohomology Vanishing;192
8.5.4;29.4 Rigidity;193
8.5.5;29.5 Chow Rings;194
8.5.6;29.6 Closures of Flats;194
8.6;30 Wonderful Varieties;195
8.6.1;30.1 Standard Completions;195
8.6.2;30.2 Demazure Embedding;197
8.6.3;30.3 Case of a Symmetric Space;198
8.6.4;30.4 Canonical Class;199
8.6.5;30.5 Cox Ring;199
8.6.6;30.6 Wonderful Varieties;203
8.6.7;30.7 How to Classify Spherical Subgroups;204
8.6.8;30.8 Spherical Spaces of Rank 1;205
8.6.9;30.9 Localization of Wonderful Varieties;207
8.6.10;30.10 Types of Simple Roots and Colors;209
8.6.11;30.11 Combinatorial Classification of Spherical Subgroups and Wonderful Varieties;210
8.6.12;30.12 Proof of the Classification Theorem;212
8.7;31 Frobenius Splitting;217
8.7.1;31.1 Basic Properties;217
8.7.2;31.2 Splitting via Differential Forms;218
8.7.3;31.3 Extension to Characteristic Zero;220
8.7.4;31.4 Spherical Case;221
9;Appendices;223
9.1;A Algebraic Geometry;223
9.1.1;A.1 Rational Singularities;223
9.1.2;A.2 Mori Theory;224
9.1.3;A.3 Schematic Points;227
9.2;B Geometric Valuations;228
9.3;C Rational Modules and Linearization;230
9.4;D Invariant Theory;232
9.5;E Hilbert Schemes;236
9.5.1;E.1 Classical Case;236
9.5.2;E.2 Nested Hilbert Scheme;238
9.5.3;E.3 Invariant Hilbert Schemes;239
10;References;242
11;Name Index;254
12;Subject Index;258
13;Notation Index;263