The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties

Abstract

We prove the following result: if a $$\,\,\,\,\,{\mathbb {Q}}\,\,\,\,\,$$ -Fano variety is uniformly K-stable, then it admits a Kähler–Einstein metric. This proves the uniform version of Yau–Tian–Donaldson conjecture for all (singular) Fano varieties with discrete automorphism groups. We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments. This perturbation approach depends on the valuative criterion and non-Archimedean estimates, and is motivated by our previous paper.