Equivariant $\mathbb R$-test configurations of polarized spherical varieties

TL;DR

This paper classifies $G$-equivariant $R$-test configurations of polarized spherical varieties using combinatorial data, providing a finiteness result for special configurations and applying it to the semistable degeneration of $Q$-Fano spherical varieties.

Contribution

It introduces a classification of $G$-equivariant $R$-test configurations for polarized spherical varieties and establishes a finiteness theorem for their central fibers.

Findings

Classified $G$-equivariant normal $R$-test configurations via combinatorial data.

Proved a finiteness theorem for central fibers of special $R$-test configurations.

Applied results to the semistable degeneration of $Q$-Fano spherical varieties.

Abstract

Let $G$ be a connected, complex reductive Lie group and $G/H$ a spherical homogenous space. Let $(X,L)$ be a polarized $G$-variety which is a spherical embedding of $G/H$. In this paper we classify $G$-equivariant normal $\mathbb R$-test configurations of $(X,L)$ via combinatory data. In particular we classify the special ones, and prove a finiteness theorem of central fibres of $G$-equivariant special $\mathbb R$-test configurations. Also, as an application we study the semistable degeneration problem of a $\mathbb Q$-Fano spherical variety.