Uniqueness properties for spherical varieties

TL;DR

This paper discusses the uniqueness of spherical varieties by examining whether certain combinatorial invariants uniquely identify varieties within classes like smooth affine, general affine, and homogeneous spaces.

Contribution

It presents results on the conditions under which combinatorial invariants uniquely determine spherical varieties in various classes.

Findings

Uniqueness of invariants for smooth affine spherical varieties

Conditions for invariants to determine general affine spherical varieties

Results on the classification of homogeneous spherical spaces

Abstract

The goal of these lectures is to explain speaker's results on uniqueness properties of spherical varieties. By a uniqueness property we mean the following. Consider some special class of spherical varieties. Define some combinatorial invariants for spherical varieties from this class. The problem is to determine whether this set of invariants specifies a spherical variety in this class uniquely (up to an isomorphism). We are interested in three classes: smooth affine varieties, general affine varieties, and homogeneous spaces.