A minimizing valuation is quasi-monomial

• 成果类型: Article
• 署名作者: Xu, Chenyang
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.191.3.6
• 发表日期: 2020
• 页码: 1003-1030
• 关键词: kahler-einstein metrics; openness conjecture; k-stability; complex; VARIETIES; EXISTENCE; IDEALS

摘要

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.