Kyeong-Dong Park, K-stability of Fano Spherical Varieties, I

The aim of this seminar is to provide participants with a comprehensive understanding of the paper “K-stability of Fano spherical varieties” by Thibaut Delcroix. For a reductive algebraic group G, a normal G-variety is called spherical if it contains an open B-orbit, where B is a fixed Borel subgroup of G. The class of spherical varieties contains several important families which were studied independently, for example, toric varieties, rational homogeneous varieties, group embeddings, horospherical varieties, symmetric varieties, and wonderful varieties. They are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties. We discuss Delcroix’s combinatorial criterion for K-stability of a smooth Fano spherical variety in terms of the barycenter of its moment polytope with respect to the Duistermaat-Heckman measure and data associated to the corresponding spherical homogeneous space.

(1) Spherical varieties, colored fans, and algebraic moment polytopes

(2) Criterion for K-stability of Fano spherical varieties and its applications

(3) Equivariant test configurations with horospherical central fiber, and modified Futaki invariant