K-stability of Q-Fano Spherical Varieties via Compatible Divisors
Renpeng Zheng
TL;DR
The paper develops a framework to compute the $G$-equivariant stability threshold for $\mathbb{Q}$-Fano spherical varieties using compatible divisors. By translating valuations into polytopal data through moment polytopes and Duistermaat–Heckman measures, it identifies a unique $G$-invariant anticanonical divisor $D^B_L$ that computes $\delta_G(L)$. The approach generalizes the toric case via compatible divisors and reduces K-stability questions to log canonical thresholds with respect to $D^B_L$. The results unify toric and spherical cases under the Abban–Zhuang philosophy and provide a practical criterion for K-(semi)stability in the spherical setting, independent of a priori linearization choices.
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.