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

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.