Acta Mathematica Sinica, English Series

, Volume 34, Issue 3, pp 371–416

# Sanya Lectures: Geometry of Spherical Varieties

Article

## 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.

## Keywords

Spherical varieties orbits local structure divisors curves

## MR(2010) Subject Classification

14M27 14-02 14L30

## Notes

### Acknowledgements

I first thank Michel Brion and Baohua Fu for the kind invitation to give lectures on the geometry of spherical varieties in Sanya. I thank Jacopo Gandini for many helpful discussions on our lectures during the two weeks of the conference. I also thank all the participants especially Johannes Hofscheier and Dmitry Timashev for the many questions and discussions during and after the talks. This led to many improvements and expansions of the first version of these notes. Finally I thank the referee for his comments and corrections.

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