On the K-stability of Fano varieties and anticanonical divisors

TL;DR

This paper explores criteria for K-stability of Fano varieties using anticanonical divisors, proposing new conditions and relating them to existing stability notions, with proofs for specific cases.

Contribution

It introduces a new sufficient condition for K-stability based on anticanonical Q-divisors and relates it to Berman-Gibbs stability, also providing an algebraic proof under Tian's alpha invariant.

Findings

Proposed a conjectural equivalence between anticanonical divisor conditions and K-stability.

Established a sufficient condition for K-stability involving anticanonical Q-divisors.

Provided an algebraic proof of K-stability for Fano varieties satisfying Tian's alpha invariant.

Abstract

We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of certain anticanonical Q-divisors of given Fano variety, which we conjecture to be equivalent to the K-stability. We prove that it is at least sufficient condition and also relate to the Berman-Gibbs stability. We also give another algebraic proof of the K-stability of Fano varieties which satisfy Tian's alpha invariants condition.