Uniform K-stability of $G$-varieties of complexity 1

TL;DR

This paper classifies test configurations of complexity 1 G-varieties, provides a formula for anti-canonical divisors, and establishes a combinatorial criterion for uniform K-stability of Q-Fano varieties.

Contribution

It introduces a combinatorial classification of G-equivariant test configurations and derives a new criterion for uniform K-stability based on this data.

Findings

Classification of G-equivariant test configurations with integral central fibre.

Formula for anti-canonical divisors on complexity 1 G-varieties.

Criterion for uniform K-stability of Q-Fano G-varieties.

Abstract

Let ${\rm k}$ be an algebraically closed field of characteristic 0 and $G$ a connect, reductive group over it. Let $X$ be a projective $G$-variety of complexity 1. We classify $G$-equivariant normal test configurations of $X$ with integral central fibre via the combinatorial data. We also give a formula of anti-canonical divisors on $X$. Based on this formula, when $X$ is $\mathbb Q$-Fano, we give an expression of the Futaki invariant, and derive a criterion of uniform K-stability in terms of the combinatorial data.