K-polystability and reduced uniform K-stability of log Fano cone singularities

TL;DR

This paper establishes the equivalence of K-polystability and reduced uniform K-stability for log Fano cone singularities, providing a key criterion for their stability properties in algebraic geometry.

Contribution

It proves that K-polystability for normal test configurations is equivalent to that for special test configurations and links reduced uniform K-stability with K-polystability.

Findings

K-polystability for normal test configurations iff for special test configurations

Reduced uniform K-stability is equivalent to K-polystability

Log Fano cones with delta-invariant ≥ 1 are K-polystable

Abstract

We prove that a log Fano cone $(X,\Delta,\xi_0)$ satisfying $\delta_\mathbb{T}(X,\Delta,\xi_0)\ge 1$ is K-polystable for normal test configurations if and only if it is K-polystable for special test configurations. We also establish the reduced uniform K-stability of $(X,\Delta,\xi_0)$ and show that it is equivalent to K-polystability.