Spherical Varieties an Introduction

Abstract

When one studies complex algebraic homogeneous spaces it is natural to begin with the ones which are complete (i.e. compact) varieties. They are the “generalized flag manifolds”. Their occurence in many problems of representation theory, algebraic geometry, … make them an important class of algebraic varieties. In order to study a noncompact homogeneous space G/H, it is equally natural to compactify it, i.e. to embed it (in a G- equivariant way) as a dense open set of a complete G-variety. A general theory of embeddings of homogeneous spaces has been developed by Luna and Vust [LV]. It works especially well in the so-called spherical case: G is reductive connected and a Borei subgroup of G has a dense orbit in G/H. (This class includes complete homogeneous spaces as well as algebraic tori and symmetric spaces). A nice feature of a spherical homogeneous space is that any embedding of it (called a spherical variety) contains only finitely many G-orbits, and these are themselves spherical. So we can hope to describe these embeddings by combinatorial invariants, and to study their geometry. I intend to present here some results and questions on the geometry (see [LV], [BLV], [BP], [Lun] for a classification of embeddings).