K-stability and polystable degenerations of polarized spherical varieties

TL;DR

This paper investigates the K-stability of polarized spherical varieties by formulating it as a variational problem involving the Futaki invariant and the homogeneous Monge-Ampère equation, especially focusing on rank two cases.

Contribution

It reduces the K-stability problem to a variational problem and characterizes minimizers via the HMA, providing explicit criteria for rank two spherical varieties.

Findings

Minimizers satisfy the homogeneous Monge-Ampère equation.

Explicit criteria for strict semistability in rank two Fano spherical varieties.

Characterization of polystable degenerations for rank two cases.

Abstract

In this paper, we study the K-stability of polarized spherical varieties. After reduction, it can be treated as a variational problem of the reduced functional of the Futaki invariant on the associated moment polytope. With the convexity constraint of the problem, the minimizers are shown to satisfy the homogeneous Monge-Amp\`ere equation (HMA). When the spherical variety has rank two, a simpler characterization can be established through properties of the HMA. As an application, we determine the strict semistability and polystable degenerations for Fano spherical varieties of rank two.