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.
Categories: math.DG math.AG math.CV