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.

From the reviews:

“The book by D. A. Timashev is a welcome survey of old and new results on actions of algebraic groups on algebraic varieties. … The book contains an extensive list of references, both to original research articles as well as to books and monographs. It is surely a welcome guide for nonexperts who want to enter the subject and a good reference for the specialist.” (Gestur Ólafsson, Bulletin of the American Mathematical Society, Vol. 51 (2), April, 2014)

“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)