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

TL;DR

This paper develops a combinatorial approach to weighted K-stability of $Q$-Fano spherical varieties, linking stability criteria with the existence of Kähler-Ricci $g$-solitons through intersection number formulas.

Contribution

It introduces a combinatorial method to compute weighted non-Archimedean functionals and establishes stability criteria for $Q$-Fano spherical varieties.

Findings

Computed weighted non-Archimedean functionals using combinatorial data.

Defined a modified Futaki invariant related to the weight g.

Established equivalence of stability notions and criteria for Kähler-Ricci g-solitons.

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 K\"ahler-Ricci $g$-solitons.