A minimizing valuation is quasi-monomial

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We prove a version of Jonsson-Mustata's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techniques to a family of klt singularities, we show that the volume of klt singularities is a constructible function. As a corollary, we prove that in a family of klt log Fano pairs, the K-semistable ones form a Zariski open set. Together with previous works by many people, we conclude that all K-semistable klt Fano varieties with a fixed dimension and volume are parametrized by an Artin stack of finite type, which then admits a separated good moduli space, whose geometric points parametrize K-polystable klt Fano varieties.

Original languageEnglish (US)
Pages (from-to)1003-1030
Number of pages28
JournalAnnals of Mathematics
Issue number3
StatePublished - May 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)


  • Complement
  • K-moduli of fano varieties
  • Local volume of klt singularities
  • Quasi-monomial


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