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Original Article

The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties

Department of Mathematics, Purdue University, West Lafayette, IN 47907-2067, USA
Present Address: Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA
School of Mathematical Sciences and BICMR, Peking University, Yiheyuan Road 5, Beijing 100871, China
School of Mathematical Sciences, Zhejiang University, Zheda Road 38, Hangzhou 310027, Zhejiang, China
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Abstract

We prove the following result: if a Q-Fano variety is uniformly K-stable, then it admits a Kähler–Einstein metric. This proves the uniform version of Yau–Tian–Donaldson conjecture for all (singular) Fano varieties with discrete automorphism groups. We achieve this by modifying Berman–Boucksom–Jonsson's strategy in the smooth case with appropriate perturbative arguments. This perturbation approach depends on the valuative criterion and non-Archimedean estimates, and is motivated by our previous paper.

Keywords

Kähler–Einstein metricsYau–Tian–Donaldson conjectureFano varietiesUniform K-stability

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Peking Mathematical Journal
Volume 5 Issue 2,
September 2022
Pages 383-426
DOI: 10.1007/s42543-021-00039-5

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Cite this article:
Li, C., Tian, G. & Wang, F. The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties. Peking Math J 5, 383-426 (2022). https://doi.org/10.1007/s42543-021-00039-5

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Received: 12 December 2020
Revised: 13 January 2021
Accepted: 22 April 2021
Published: 02 August 2021
© Peking University 2021