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

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Abstract

Let X be any Q-Fano variety and Aut (X) be the identity component of the automorphism group of X. Let G be a connected reductive subgroup of Aut (X) that contains a maximal torus of Aut (X). We prove that X admits a Kähler–Einstein metric if and only if X is 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 G-uniform K-stability.

Original languageEnglish (US)
Pages (from-to)661-744
Number of pages84
JournalInventiones Mathematicae
Volume227
Issue number2
DOIs
StatePublished - Feb 2022

All Science Journal Classification (ASJC) codes

  • General Mathematics

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