Existence of valuations with smallest normalized volume

TL;DR

This paper proves Li's conjecture that a valuation with the smallest normalized volume exists over any klt singularity, and shows such a valuation need not be divisorial, advancing understanding of K-stability.

Contribution

The paper confirms the existence of minimal normalized volume valuations over klt singularities and provides a counterexample to divisoriality of such valuations.

Findings

Existence of valuations with smallest normalized volume over klt singularities

Counterexample showing minimal valuation need not be divisorial

Advancement in understanding of K-stability and valuation theory

Abstract

Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a klt singularity there exists a valuation with smallest normalized volume. We prove this conjecture and provide an example to show that such a valuation need not be divisorial.