A Minimizing Valuation is Quasi-monomial

Chenyang Xu

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arXiv · 1907.01114

A Minimizing Valuation is Quasi-monomial

Chenyang Xu

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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 [Jia17], 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 by [BX18, ABHLX19], whose geometric points parametrize K-polystable klt Fano varieties.

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Chenyang Xu. 2019-11-18. A Minimizing Valuation is Quasi-monomial. https://arxiv.org/abs/1907.01114

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