On the semi-continuity problem of normalized volumes of singularities

TL;DR

This paper investigates the behavior of normalized volumes in families of klt singularities, showing they only decrease at countably many subvarieties, and explores the genericity of K-semistability in log Fano pairs.

Contribution

It establishes semi-continuity properties of normalized volumes and demonstrates the generic nature of K-semistability in certain algebraic families.

Findings

Normalized volumes only jump down at countably many subvarieties.

Smooth points have the largest normalized volume among klt singularities.

K-semistability is very generic or empty in families of log Fano pairs.

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.