Stable degeneration of families of klt singularities with constant local volume

TL;DR

This paper demonstrates that in stable families of klt singularities with constant local volume, the associated valuations form a flat family of ideals leading to a degeneration into K-semistable log Fano cone singularities.

Contribution

It extends the method of Xu and Zhuang to families, showing how valuations and degenerations behave in stable klt singularity families with constant local volume.

Findings

Valuations form a flat family of ideals.

Degeneration to K-semistable log Fano cones.

Supports stability and finite generation in families.

Abstract

We prove that for a locally stable family of klt singularities with constant local volume, the ideal sequences of the minimizing valuations for the normalized volume function form a family of ideals with flat cosupport, which induces a degeneration to a locally stable family of K-semistable log Fano cone singularities. Our proof is a family version of the method of C. Xu and Z. Zhuang proving finite generation by Koll\'ar models and multiple degenerations.