A minimizing valuation is quasi-monomial

Abstract

We prove a version of Jonsson-Mustaţǎ’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.

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      volume = {42},
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      number = {5},
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      title = {Special test configuration and {K}-stability of {F}ano varieties},
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Authors

Chenyang Xu

Massachusetts Institute of Technology, Cambridge, MA and BICMR, Peking University, Beijing, China