A note on Koll\'{a}r valuations

TL;DR

This paper investigates the structure of Kollár valuations within the dual complex of klt singularities, proving their path connectivity and classifying cases with one-dimensional dual complexes.

Contribution

It establishes the path connectedness of Kollár valuations in the dual complex and classifies the one-dimensional cases, advancing understanding of singularity valuations.

Findings

Kollár valuations form a path connected set in the dual complex.

Classification of dual complexes that are one-dimensional.

Enhanced understanding of valuation structures in klt singularities.

Abstract

We prove the set of Koll\'{a}r valuations in the dual complex of a klt singularity with a fixed complement is path connected. We also classify the case when the dual complex is one dimensional.