On the semi-continuity problem of normalized volumes of singularities

Abstract

We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes can only jump down at countably many subvarieties. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed by Li, Xu and the author, we show that K-semistability is a very generic or empty condition in any $\mathbb{Q}$-Gorenstein flat family of log Fano pairs.

16 pages. Comments are very welcome

References in corpus (2)

• Finiteness of étale fundamental groups by reduction modulo $p$
• Seifert $G_m$-bundles

Cited by in corpus (1)

• Interaction Between Singularity Theory and the Minimal Model Program