K-stability of Q-Fano Spherical Varieties via Compatible Divisors

TL;DR

This paper investigates the K-stability of ${Q}$-Fano spherical varieties with reductive group actions, identifying a unique invariant divisor that determines their stability status, thus advancing understanding of their geometric properties.

Contribution

It introduces a method to determine K-stability of ${Q}$-Fano spherical varieties via a unique invariant divisor associated with compatible divisors.

Findings

Existence of a unique anticanonical ${Q}$-divisor computing the stability threshold.

The invariant divisor characterizes the K-stability of the variety.

The divisor is invariant under the Borel subgroup action.

Abstract

We study the K-stability of $\mathbb{Q}$-Fano spherical varieties using compatible divisors. More precisely, if the $\mathbb{Q}$-Fano variety, with a reductive group action, has an open Borel subgroup orbit, then there is a unique anticanonical $\mathbb{Q}$-divisor computing the equivariant stability threshold. This $\mathbb{Q}$-divisor is invariant under the Borel subgroup action, and it characterizes the K-stability of a $\mathbb{Q}$-Fano spherical variety.