A Minimizing Valuation is Quasi-monomial

TL;DR

This paper proves that for graded sequences of ideals, a quasi-monomial valuation computes the log canonical threshold, confirms a conjecture about minimizers of normalized volume being quasi-monomial, and explores properties of klt singularities and Fano varieties.

Contribution

It establishes the existence of quasi-monomial valuations computing log canonical thresholds and confirms that minimizers of normalized volume are always quasi-monomial, advancing singularity theory.

Findings

Volume of klt singularities is a constructible function.

K-semistable klt Fano pairs form a Zariski open set.

Parametrization of K-semistable klt Fano varieties by an Artin stack.

Abstract

We prove a version of Jonsson-Musta\c{t}\v{a}'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.