Alpha invariants and K-stability for general polarisations of Fano varieties

TL;DR

This paper establishes a new criterion for K-stability of Fano varieties with general polarisations using Tian's alpha invariant, extending previous results and providing new stable examples.

Contribution

It generalizes Odaka-Sano's result to broader polarisations and introduces a sufficient condition for K-stability based on the alpha invariant.

Findings

Provides new K-stable polarisations for degree one del Pezzo surfaces

Extends K-stability criteria to general polarisations of Fano varieties

Proves a similar criterion for log K-stability

Abstract

We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian's alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system associated to the polarisation. This generalises a result of Odaka-Sano in the anti-canonically polarised case, which is the algebraic counterpart of Tian's analytic criterion implying the existence of a K\"{a}hler-Einstein metric. As an application, we give new K-stable polarisations of a general degree one del Pezzo surface. We also prove a corresponding result for log K-stability.