G-uniform stability and K\"{a}hler-Einstein metrics on Fano varieties

TL;DR

This paper establishes that a $Q$-Fano variety admits a K"ahler-Einstein metric if and only if it is $G$-uniformly K-stable, confirming a version of the Yau-Tian-Donaldson conjecture for singular Fano varieties, using a new valuative criterion.

Contribution

It proves the equivalence between the existence of K"ahler-Einstein metrics and $G$-uniform K-stability for $Q$-Fano varieties, extending the Yau-Tian-Donaldson conjecture to singular cases with a novel valuative criterion.

Findings

Equivalence between K"ahler-Einstein metrics and $G$-uniform K-stability.

Validation of the Yau-Tian-Donaldson conjecture for singular Fano varieties.

Introduction of a valuative criterion for $G$-uniform K-stability.

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 K\"{a}hler-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.