Homogeneous spaces and equivariant embeddings / Dmitry A. Timashev.

Format:

Book

Author/Creator:

Timashev, Dmitry A.

Series:

Encyclopaedia of mathematical sciences ; 138.

Encyclopaedia of mathematical sciences. Invariant theory and algebraic transformation groups ; 8.

Encyclopaedia of mathematical sciences ; 138

Encyclopaedia of mathematical sciences. Invariant theory and algebraic transformation groups ; 8

Language:

English

Subjects (All):

Homogeneous spaces.

Physical Description:

xxi, 253 pages ; 24 cm.

Place of Publication:

Berlin : Springer, 2011.

Summary:

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. Book jacket.

Contents:

1 Algebraic Homogeneous Spaces 1

1 Homogeneous Spaces 1

1.1 Basic Definitions 1

1.2 Tangent Spaces and Automorphisms 3

2 Fibrations, Bundles, and Representations 3

2.1 Homogeneous Bundles 3

2.2 Induction and Restriction 5

2.3 Multiplicities 6

2.4 Regular Representation 6

2.5 Hecke Algebras 7

2.6 Weyl Modules 8

3 Classes of Homogeneous Spaces 10

3.1 Reductions 10

3.2 Projective Homogeneous Spaces 11

3.3 Affine Homogeneous Spaces 11

3.4 Quasiaffine Homogeneous Spaces 13

2 Complexity and Rank 15

4 Local Structure Theorems 15

4.1 Locally Linearizable Actions 15

4.2 Local Structure of an Action 16

4.3 Local Structure Theorem of Knop 19

5 Complexity and Rank of G-varieties 20

5.1 Basic Definitions 20

5.2 Complexity and Rank of Subvarieties 20

5.3 Weight Semigroup 22

5.4 Complexity and Growth of Multiplicities 22

6 Complexity and Modality 24

6.1 Modality of an Action 24

6.2 Complexity and B-modality 25

6.3 Adherence of B-orbits 26

6.4 Complexity and G-modality 27

7 Horospherical Varieties 28

7.1 Horospherical Subgroups and Varieties 28

7.2 Horospherical Type 30

7.3 Horospherical Contraction 30

8 Geometry of Cotangent Bundles 31

8.1 Symplectic Structure 31

8.2 Moment Map 31

8.3 Localization 32

8.4 Logarithmic Version 33

8.5 Image of the Moment Map 33

8.6 Corank and Defect 35

8.7 Cotangent Bundle and Geometry of an Action 36

8.8 Doubled Actions 37

9 Complexity and Rank of Homogeneous Spaces 39

9.1 Gener'l Formul'e 39

9.2 Reduction to Representations 41

10 Spaces of Small Rank and Complexity 43

10.1 Spaces of Rank ≤ 1 43

10.2 Spaces of Complexity ≤ 1 44

11 Double Cones 46

11.1 HV-cones and Double Cones 47

11.2 Complexity and Rank 49

11.3 Factorial Double Cones of Complexity ≤ 1 51

11.4 Applications to Representation Theory 52

11.5 Spherical Double Cones 55

3 General Theory of Embeddings 57

12 The Luna-Vust Theory 57

12.1 Equivariant Classification of G-varieties 57

12.2 Universal Model 58

12.3 Germs of Subvarieties 60

12.4 Morphisms, Separation, and Properness 61

13 B-charts 62

13.1 B-charts and Colored Equipment 62

13.2 Colored Data 63

13.3 Local Structure 65

14 Classification of G-models 66

14.1 G-germs 66

14.2 G-models 67

15 Case of Complexity 0 68

15.1 Combinatorial Description of Spherical Varieties 68

15.2 Functoriality 70

15.3 Orbits and Local Geometry 71

16 Case of Complexity 1 72

16.1 Generically Transitive and One-parametric Cases 72

16.2 Hyperspace 73

16.3 Hypercones 75

16.4 Colored Data 77

16.5 Examples 80

16.6 Local Properties 84

17 Divisors 84

17.1 Reduction to B-stable Divisors 84

17.2 Cartier Divisors 85

17.3 Case of Complexity ≤ 1 86

17.4 Global Sections of Line Bundles 89

17.5 Ample Divisors 91

18 Intersection Theory 94

18.1 Reduction to B-stable Cycles 94

18.2 Intersection of Divisors 95

18.3 Divisors and Curves 99

18.4 Chow Rings 100

18.5 Halphen Ring 101

18.6 Generalization of the Bézout Theorem 102

4 Invariant Valuations 105

19 G-valuations 106

19.1 Basic Properties 106

19.2 Case of a Reductive Group 107

20 Valuation Cones 108

20.1 Hyperspace 108

20.2 Main Theorem 110

20.3 A Good G-model 110

20.4 Criterion of Geometricity 111

20.5 Proof of the Main Theorem 112

20.6 Parabolic Induction 114

21 Central Valuations 115

21.1 Central Valuation Cone 115

21.2 Central Automorphisms 116

21.3 Valuative Characterization of Horospherical Varieties 118

21.4 G-valuations of a Central Divisor 118

22 Little Weyl Group 119

22.1 Normalized Moment Map 119

22.2 Conormal Bundle to General U-orbits 120

22.3 Little Weyl Group 121

22.4 Relation to Valuation Cones 123

23 Invariant Collective Motion 124

23.1 Polarized Cotangent Bundle 124

23.2 Integration of Invariant Collective Motion 125

23.3 Flats and Their Closures 126

23.4 Non-symplectically Stable Case 128

23.5 Proof of Theorem 22.13 129

23.6 Sources 130

23.7 Root System of a G-variety 131

24 Formal Curves 132

24.1 Valuations via Germs of Curves 132

24.2 Valuations via Formal Curves 133

5 Spherical Varieties 135

25 Various Characterizations of Sphericity 136

25.1 Spherical Spaces 136

25.2 "Multiplicity-free" Property 137

25.3 Weakly Symmetric Spaces and Gelfand Pairs 138

25.4 Commutativity 139

25.5 Generalizations 142

26 Symmetric Spaces 145

26.1 Algebraic Symmetric Spaces 145

26.2 θ-stable Tori 146

26.3 Maximal θ-fixed Tori 147

26.4 Maximal θ-split Tori 148

26.5 Classification 150

26.6 Weyl Group 154

26.7 B-orbits 154

26.8 Colored Equipment 155

26.9 Coisotropy Representation 157

26.10 Flats 157

27 Algebraic Monoids and Group Embeddings 158

27.1 Algebraic Monoids 158

27.2 Reductive Monoids 160

27.3 Orbits 162

27.4 Normality and Smoothness 164

27.5 Group Embeddings 165

27.6 Enveloping and Asymptotic Semigroups 168

28 S-varieties 169

28.1 General S-varieties 169

28.2 Affine Case 170

28.3 Smoothness 173

29 Toroidal Embeddings 173

29.1 Toroidal Versus Toric Varieties 174

29.2 Smooth Toroidal Varieties 174

29.3 Cohomology Vanishing 176

29.4 Rigidity 177

29.5 Chow Rings 178

29.6 Closures of Flats 178

30 Wonderful Varieties 179

30.1 Standard Completions 179

30.2 Demazure Embedding 181

30.3 Case of a Symmetric Space 182

30.4 Canonical Class 183

30.5 Cox Ring 183

30.6 Wonderful Varieties 187

30.7 How to Classify Spherical Subgroups 188

30.8 Spherical Spaces of Rank 1 189

30.9 Localization of Wonderful Varieties 191

30.10 Types of Simple Roots and Colors 193

30.11 Combinatorial Classification of Spherical Subgroups and Wonderful Varieties 194

30.12 Proof of the Classification Theorem 196

31 Frobenius Splitting 201

31.1 Basic Properties 201

31.2 Splitting via Differential Forms 202

31.3 Extension to Characteristic Zero 204

31.4 Spherical Case 205.

Notes:

Includes bibliographical references and indexes.

ISBN:

3642183980

9783642183980

OCLC:

706784521