Homogeneous Spaces and Equivariant Embeddings [electronic resource] / by D.A. Timashev.
Contributor(s): SpringerLink (Online service)Material type: TextSeries: Encyclopaedia of Mathematical Sciences: 138Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2011Description: XXII, 254 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783642183997Subject(s): Mathematics | Algebraic geometry | Topological groups | Lie groups | Mathematics | Algebraic Geometry | Topological Groups, Lie GroupsAdditional physical formats: Printed edition:: No titleDDC classification: 516.35 LOC classification: QA564-609Online resources: Click here to access online
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Introduction.- 1 Algebraic Homogeneous Spaces -- 2 Complexity and Rank -- 3 General Theory of Embeddings -- 4 Invariant Valuations -- 5 Spherical Varieties -- Appendices -- Bibliography -- Indices.
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.