K-Stability of Fano spherical varieties

TL;DR

This paper establishes a combinatorial criterion for K-stability of Fano spherical varieties, linking geometric stability to moment polytopes and combinatorial data, and applies it to Kähler-Einstein metrics and Kähler-Ricci solitons.

Contribution

It provides a new criterion for K-stability of Fano spherical varieties based on their moment polytopes and combinatorial data, extending to Kähler-Ricci solitons.

Findings

Criterion for K-stability using moment polytope and combinatorial data.

Application to existence of Kähler-Einstein metrics on spherical Fano manifolds.

Extension to modified K-stability and Kähler-Ricci solitons.

Abstract

We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds proved by Datar and Sz\'ekelyhidi, it yields a criterion for the existence of a K\"ahler-Einstein metric on a spherical Fano manifold. The results hold also for modified K-stability and existence of K\"ahler-Ricci solitons.