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

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 Kahler–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.