Space of Filtrations and Singularities

Abstract

In this thesis, we systematically study the structure of the space of saturated filtrationson a Noetherian local domain, and consider its application in commutative algebra, K-stability and birational geometry. Introduced in the joint work with Harold Blum and Yuchen Liu, saturation can be used to characterize when two filtrations have equal multiplicities, generalizing Rees’ theorem concerning Hilbert-Samuel multiplicity and integral closure of ideals. Next, we present the joint work with Blum and Liu on the convexity of multiplicities along geodesics. This result allows us to prove the convexity of volumes on a single simplex in the valuation space, generalizing the result of Boucksom, Favre and Jonsson. Moreover, it gives a new proof to the uniqueness of normalized volume minimizers originally proven by Xu and Zhuang, which is part of the Stable Degeneration Conjecture. In order to justify the term geodesic, we introduce a metric d1, mimicking the Darvas metric in complex geometry and global non-archimedean geometry. The constructions endow the space of saturated filtrations with the structure of a geodesic metric space. We also define several other topologies on the space. As an application, we show some continuity properties of the log canonical threshold function, in the spirit of Kollár-Demailly.