Sanya lectures : geometry of spherical varieties

TL;DR

This paper reviews the geometry of spherical varieties, focusing on their structure, divisors, and curves, providing explicit descriptions and conditions for smoothness, based on lecture notes from Sanya.

Contribution

It offers a comprehensive overview of the structure and geometry of spherical varieties, including explicit descriptions of divisors, curves, and smoothness conditions.

Findings

Description of B-orbits and local structure theorems

Explicit B-stable canonical divisor construction

Conditions for smoothness of spherical varieties

Abstract

These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of $B$-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider $B$-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit $B$-stable canonical divisor on any spherical variety.