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

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¨ahler-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.Supported by a studentship associated to an EPSRC Career Acceleration Fellowship (EP/J002062/1)This is the author accepted manuscript. The final version is available from Oxford University Press via http://dx.doi.org/10.1093/imrn/rnu16