Weighted K-stability of $\mathbb{Q}$-Fano spherical varieties

Abstract

Let $G$ be a connected, complex reductive Lie group and $X$ a $\mathbb{Q}$-Fano $G$-spherical variety. In this paper we compute the weighed non-Archimedean functionals of a $G$-equivariant normal test configurations of $X$ via combinatory data. Also we define a modified Futaki invariant with respect to the weight $g$, and give an expression in terms of intersection numbers. Finally we show the equivalence of different notations of stability and gives a stability criterion on $\mathbb{Q}$-Fano spherical varieties, which is also a criterion of existence of Kähler-Ricci $g$-solitons.

We added sections on weighted NA functionals and modified Futaki invariants on a general $\mathbb{Q}$-Fano variety