Invariant theory and algebraic transformation groups. Vol. 8 (9) : Homogeneous spaces and equivariant embeddings

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. 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 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 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 in terms of 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 tried to cover all most important issues, including a substantial progress obtained in the theory of spherical varieties and around it quite recently Front Matter....Pages I-XXI Algebraic Homogeneous Spaces....Pages 1-14 Complexity and Rank....Pages 15-55 General Theory of Embeddings....Pages 57-103 Invariant Valuations....Pages 105-134 Spherical Varieties....Pages 135-206 Back Matter....Pages 207-253 Algebraic Homogeneous Spaces -- Complexity And Rank -- General Theory Of Embedding -- Invariant Valuations -- Spherical Varieties. Dmitry A. Timashev. Includes Bibliographical References (p. 227-238) And Indexes.