G-uniform stability and Kähler–Einstein metrics on Fano varieties
2021
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.
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