G-uniform stability and Kähler-Einstein metrics on Fano varieties

Abstract

Let $X$ be any $\mathbb{Q}$-Fano variety and $\mathrm{Aut}(X{)}_0$ be the identity component of the automorphism group of $X$. Let $\mathbb{G}$ be a connected reductive subgroup of $\mathrm{Aut}(X{)}_0$ that contains a maximal torus of $\mathrm{Aut}(X{)}_0$. We prove that $X$ admits a Kähler-Einstein metric if and only if $X$ is $\mathbb{G}$-uniformly K-stable. This proves a version of Yau-Tian-Donaldson conjecture for arbitrary singular Fano varieties. A key new ingredient is a valuative criterion for $\mathbb{G}$-uniform K-stability.

60 pages. Thoroughly revised and the presentation improved. Accepted by Invent. Math

Cited by in corpus (1)

• On K-stability of some del Pezzo surfaces of Fano index 2