The normalized volume of a singularity is lower semicontinuous

TL;DR

This paper proves that normalized volumes of klt singularities are lower semicontinuous in families, implying smooth points maximize normalized volume, and explores the genericity of K-semistability in log Fano pairs.

Contribution

It establishes the lower semicontinuity of normalized volumes in families of klt singularities and links K-semistability to generic conditions in such families.

Findings

Normalized volumes are lower semicontinuous in families.

Smooth points have the largest normalized volume among klt singularities.

K-semistability is a very generic or empty condition in certain families.

Abstract

We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. 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, Liu and Xu, we show that K-semistability is a very generic or empty condition in any $\mathbb{Q}$-Gorenstein flat family of log Fano pairs.