Alpha Invariants and K-Stability for General Polarizations of Fano Varieties

Abstract

We provide a sufficient condition for polarizations 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 polarization. This generalizes a result of Odaka–Sano in the anti-canonically polarized case, which is the algebraic counterpart of Tian's analytic criterion implying the existence of a Kähler–Einstein metric. As an application, we give new K-stable polarizations of a general degree 1 del Pezzo surface. We also prove a corresponding result for log K-stability.