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

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Abstract

We prove the following result: if a \(\,\,\,\,\,{\mathbb {Q}}\,\,\,\,\,\)-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.

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Acknowledgements

C. Li is partially supported by NSF (Grant No. DMS-1810867) and an Alfred P. Sloan research fellowship. G. Tian is partially supported by NSF (Grant No. DMS-1607091) and NSFC (Grant No. 11331001). F. Wang is partially supported by NSFC (Grant No. 11501501). The first author would like to thank S. Boucksom, M. Jonsson and L. Lempert for helpful conversations, and Y. Liu, C. Xu and M. Xia for useful comments. We would like to thank R. Berman, T. Darvas for communications that help our proof of the convexity of Mabuchi energy, and Di Nezza and V. Guedj for clarifications on regularity of geodesics. We would also like to thank anonymous referees for helpful suggestions on improving the paper.

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  1. Chi Li

    Present address: Department of Mathematics, Rutgers University, Piscataway, NJ, 08854-8019, USA

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-2067, USA

    Chi Li

  2. School of Mathematical Sciences and BICMR, Peking University, Yiheyuan Road 5, Beijing, 100871, China

    Gang Tian

  3. School of Mathematical Sciences, Zhejiang University, Zheda Road 38, Hangzhou, 310027, Zhejiang, China

    Feng Wang

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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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