Finite generation for valuations computing stability thresholds and applications to K-stability

Yuchen Liu; Chenyang Xu; Ziquan Zhuang

doi:10.4007/ANNALS.2022.196.2.2

Finite generation for valuations computing stability thresholds and applications to K-stability

Yuchen Liu^*, Chenyang Xu, Ziquan Zhuang

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

39 Scopus citations

Abstract

We prove that on any log Fano pair of dimension n whose stability threshold is less than (Formula Presented), 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 K-ahler-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.

Original language	English (US)
Pages (from-to)	507-566
Number of pages	60
Journal	Annals of Mathematics
Volume	196
Issue number	2
DOIs	https://doi.org/10.4007/ANNALS.2022.196.2.2
State	Published - Sep 2022

Funding

Acknowledgements. We would like to thank Harold Blum, Xiaowei Wang and Chuyu Zhou for helpful discussions. We also would like to thank the anonymous referees for many helpful comments. YL is partially supported by NSF Grant DMS-2148266 (formerly DMS-2001317). CX is partially supported by NSF Grant DMS-1901849 and DMS-1952531. ZZ is partially supported by NSF Grant DMS-2055531.

Keywords

Fano variety
Higher rank finite generation
K-ahler{einstein metric
K-moduli
K-stability

ASJC Scopus subject areas

Mathematics (miscellaneous)

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10.4007/ANNALS.2022.196.2.2

Cite this

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title = "Finite generation for valuations computing stability thresholds and applications to K-stability",

abstract = "We prove that on any log Fano pair of dimension n whose stability threshold is less than (Formula Presented), 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 K-ahler-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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author = "Yuchen Liu and Chenyang Xu and Ziquan Zhuang",

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AU - Xu, Chenyang

AU - Zhuang, Ziquan

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N2 - We prove that on any log Fano pair of dimension n whose stability threshold is less than (Formula Presented), 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 K-ahler-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.

AB - We prove that on any log Fano pair of dimension n whose stability threshold is less than (Formula Presented), 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 K-ahler-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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KW - K-ahler{einstein metric

KW - K-moduli

KW - K-stability

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Finite generation for valuations computing stability thresholds and applications to K-stability

Abstract

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Keywords

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