Uniform K-stability of polarized spherical varieties

TL;DR

This paper characterizes K-stability of polarized spherical varieties using combinatorial data, providing explicit criteria for the existence of constant scalar curvature Kähler metrics and relating different stability notions.

Contribution

It generalizes K-stability criteria from toric to spherical varieties and offers a combinatorial sufficient condition for G-uniform K-stability.

Findings

Provides a combinatorial criterion for G-uniform K-stability.

Shows equivalence of G-uniform K-stability and K-polystability in certain cases.

Offers an explicit checkable condition for constant scalar curvature Kähler metrics.

Abstract

We express notions of K-stability of polarized spherical varieties in terms of combinatorial data, vastly generalizing the case of toric varieties. We then provide a combinatorial sufficient condition of G-uniform K-stability by studying the corresponding convex geometric problem. Thanks to recent work of Chi Li and a remark by Yuji Odaka, this provides an explicitly checkable sufficient condition of existence of constant scalar curvature Kahler metrics. As a side effect, we show that, on several families of spherical varieties, G-uniform K-stability is equivalent to K-polystability with respect to G-equivariant test configurations for polarizations close to the anticanonical bundle.