On Berman-Gibbs stability and K-stability of $\mathbb{Q}$-Fano varieties

TL;DR

This paper establishes a link between Berman-Gibbs stability and K-stability for $Q$-Fano varieties, showing that Berman-Gibbs stability implies K-stability.

Contribution

It proves that Berman-Gibbs stability of $Q$-Fano varieties guarantees their K-stability, clarifying the relationship between these stability notions.

Findings

Berman-Gibbs stable varieties are K-stable.

Berman-Gibbs semistable varieties are K-semistable.

The paper extends the understanding of stability conditions in algebraic geometry.

Abstract

The notion of Berman-Gibbs stability was originally introduced by Robert Berman for $\mathbb{Q}$-Fano varieties $X$. We show that the pair $(X, -K_X)$ is K-stable (resp. K-semistable) provided that $X$ is Berman-Gibbs stable (resp. semistable).