Stability of Valuations and Koll\'ar Components

TL;DR

This paper establishes the uniqueness of K-semistable Kollár components among plt blow-ups of klt singularities, linking normalized volume minimization to K-semistability and showing approximation of normalized volumes by Kollár components.

Contribution

It proves the uniqueness of K-semistable Kollár components and connects normalized volume minimization with K-semistability in klt singularities.

Findings

At most one Kollár component is K-semistable among all plt blow-ups.

Normalized volume minimizers correspond to K-semistable Kollár components.

Normalized volumes of Kollár components approximate the infimum of the normalized volume function.

Abstract

We prove that among all Koll\'ar components obtained by plt blow ups of a klt singularity $o \in (X, D)$, there is at most one that is (log-)K-semistable. We achieve this by showing that if such a Koll\'ar component exists, it uniquely minimizes the normalized volume function introduced in [Li15a] among all divisorial valuations. Conversely, we show any divisorial minimizer of the normalized volume function yields a K-semistable Koll\'ar component. We also prove that for any klt singularity, the infimum of the normalized function is always approximated by the normalized volumes of Koll\'ar components.