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

Pages 507-566 from Volume 196 (2022), Issue 2 by Yuchen Liu, Chenyang Xu, Ziquan Zhuang

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 $\mathrm {K}$-stable (resp. reduced uniformly $\mathrm {K}$-stable) if and only if it is $\mathrm {K}$-stable (resp. $\mathrm {K}$-polystable); (b) the $\mathrm {K}$-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of Kähler-Einstein metric and reduced uniform $\mathrm {K}$-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.

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      title = {Gromov-{H}ausdorff limits of {K}ähler manifolds and algebraic geometry},
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      mrclass = {53C55 (32Q20 53C23)},
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      mrreviewer = {Valentino Tosatti},
      doi = {10.1007/s11511-014-0116-3},
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      journal = {Amer. J. Math.},
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      author = {de Fernex, Tommaso and Koll{\'{a}}r, J{\'{a}}nos and Xu, Chenyang},
      title = {The dual complex of singularities},
      booktitle = {Higher Dimensional Algebraic Geometry---In Honour of {P}rofessor {Y}ujiro {K}awamata's Sixtieth Birthday},
      series = {Adv. Stud. Pure Math.},
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      publisher = {Math. Soc. Japan, Tokyo},
      year = {2017},
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      title = {A valuative criterion for uniform {K}-stability of {$\Bbb Q$}-{F}ano varieties},
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      year = {2000},
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      journal = {Peking Math. J.},
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      year = {2021},
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      journal = {Duke Math. J.},
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      author = {Li, Chi and Wang, Xiaowei and Xu, Chenyang},
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      journal = {J. Amer. Math. Soc.},
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      volume = {34},
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      mrclass = {14B07 (14E30 14J17 14J45 53C55)},
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      fjournal = {Annals of Mathematics. Second Series},
      volume = {180},
      year = {2014},
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      mrclass = {14J45 (14E30 14J10 14J80)},
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      mrreviewer = {Anne-Sophie Kaloghiros},
      doi = {10.4007/annals.2014.180.1.4},
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      title = {Stability of valuations: higher rational rank},
      journal = {Peking Math. J.},
      fjournal = {Peking Mathematical Journal},
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      year = {2018},
      number = {1},
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      TITLE = {On the sharpness of {T}ian's criterion for {K}-stability},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {245},
      YEAR = {2022},
      PAGES = {41--73},
      ISSN = {0027-7630},
      MRCLASS = {14J45 (14L24 32Q26)},
      MRNUMBER = {4413362},
      DOI = {10.1017/nmj.2020.28},
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Keywords
, , , ,
DOI
https://doi.org/10.4007/annals.2022.196.2.2
MR
4445441
zbMATH
7583010
Milestones
Received: 22 February 2021
Revised: 10 February 2022
Accepted: 11 February 2022
Published online: 28 June 2022

Authors

Yuchen Liu

Northwestern University, Evanston, IL, USA

Chenyang Xu

Beijing International Center for Mathematical Research, Beijing, China

Current address:

Princeton University, Princeton, NJ, USA Ziquan Zhuang

Massachusetts Institute of Technology, Cambridge, MA