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

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.

39 pages. v2: Section 5 expanded, proof of Theorem 1.4 modified, and an appendix on local Okounkov bodies added