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September 2022 Finite generation for valuations computing stability thresholds and applications to K-stability
Yuchen Liu, Chenyang Xu, Ziquan Zhuang
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Ann. of Math. (2) 196(2): 507-566 (September 2022). DOI: 10.4007/annals.2022.196.2.2

Abstract

We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of Kähler-Einstein metric and reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.

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Yuchen Liu. Chenyang Xu. Ziquan Zhuang. "Finite generation for valuations computing stability thresholds and applications to K-stability." Ann. of Math. (2) 196 (2) 507 - 566, September 2022. https://doi.org/10.4007/annals.2022.196.2.2

Information

Published: September 2022
First available in Project Euclid: 28 June 2022

Digital Object Identifier: 10.4007/annals.2022.196.2.2

Subjects:
Primary: 14D20 , 14E99 , 14J45 , 32Q20

Keywords: {K}-moduli , {K}-stability , Fano variety , Higher Rank Finite Generation , K___hler--Einstein metric

Rights: Copyright © 2022 Department of Mathematics, Princeton University

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Vol.196 • No. 2 • September 2022
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