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.
"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