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

Yuchen Liu, Chenyang Xu, Ziquan Zhuang

Research output: Contribution to journalArticlepeer-review

1 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 languageEnglish (US)
Pages (from-to)507-566
Number of pages60
JournalAnnals of Mathematics
Volume196
Issue number2
DOIs
StatePublished - Sep 2022
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)

Keywords

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

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