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May 2020 A minimizing valuation is quasi-monomial
Chenyang Xu
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Ann. of Math. (2) 191(3): 1003-1030 (May 2020). DOI: 10.4007/annals.2020.191.3.6

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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Chenyang Xu. "A minimizing valuation is quasi-monomial." Ann. of Math. (2) 191 (3) 1003 - 1030, May 2020. https://doi.org/10.4007/annals.2020.191.3.6

Information

Published: May 2020
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2020.191.3.6

Subjects:
Primary: 14E30 , 14J17 , 14J45

Keywords: $K$-moduli of Fano varieties , complement , local volume of klt singularities , quasi-monomial

Rights: Copyright © 2020 Department of Mathematics, Princeton University

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Vol.191 • No. 3 • May 2020
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