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 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.