K-semistability of log Fano cone singularities

TL;DR

This paper provides a non-Archimedean criterion for K-semistability of log Fano cone singularities, aligning with previous definitions, and simplifies testing by focusing on special test configurations.

Contribution

It introduces a non-Archimedean perspective on K-semistability and demonstrates that testing special configurations suffices, connecting to lc places of bounded complements.

Findings

Non-Archimedean characterization matches original definition

Testing special test configurations is sufficient for K-semistability

Special test configurations relate to lc places of bounded complements

Abstract

We give a non-Archimedean characterization of K-semistability of log Fano cone singularities, and show that it agrees with the definition originally defined by Collins--Sz\'ekelyhidi. As an application, we show that to test K-semistability, it suffices to test special test configurations. We also show that special test configurations give rise to lc places of torus equivariant bounded complements.