Finite generation for valuations computing stability thresholds and applications to K-stability

Abstract

We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac {n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly $\mathrm {K}$-stable (resp. reduced uniformly $\mathrm {K}$-stable) if and only if it is $\mathrm {K}$-stable (resp. $\mathrm {K}$-polystable); (b) the $\mathrm {K}$-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of Kähler-Einstein metric and reduced uniform $\mathrm {K}$-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.