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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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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DOI: https://doi.org/10.1007/s42543-021-00039-5