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Timashev, D. A.

Homogeneous Spaces and Equivariant Embeddings (Gebundene Ausgabe)

Reihe: Encyclopaedia of Mathematical Sciences 138

Springer-Verlag GmbH, Springer Berlin, Mai 2011

Edition - 253 S. - Sprache: Englisch - 16 schwarz-weiße Abbildungen, 11 schwarz-weiße Tabellen - 245x167x22 mm Book

ISBN: 3642183980 EAN: 9783642183980

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.

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First existing complete survey of various topics related to the subject of the book and to each other, which were by now scattered in the literature. Well-developed cross-referencing mechanism makes it easy to navigate through the text, so that it may serve well as a kind of a guide or encyclopaedia on the subject, with many indications for further reading. Proofs of all important results are provided unless they are too long or technical, so that the reader can learn the theory in general from the very beginning with no special need to address to other sources.


Introduction.- 1 Algebraic Homogeneous Spaces.- 2 Complexity and Rank.- 3 General Theory of Embeddings.- 4 Invariant Valuations.- 5 Spherical Varieties.- Appendices.- Bibliography.- Indices


From the reviews:"This book is a survey of the study of equivariant embeddings of homogeneous spaces for connected reductive algebraic groups. ... The book is well written and puts together many results in a complete and concise manner." (Lucy Moser-Jauslin, Mathematical Reviews, Issue 2012 e)"The book under review is concerned with the study of equivariant embeddings of homogeneous spaces ... . This monograph provides a compact survey of these issues, including many results of Brion, Knop, Luna, and Vust, among others, as well as the author's own significant contributions to the field. ... The author admirably fills a gap in the current literature on algebraic groups with this book. It should make a welcome addition to any mathematician's library." (Nathan Ilten, Zentralblatt MATH, Vol. 1237, 2012)

ISBN 3-642-18398-0

ISBN 978-3-642-18398-0

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