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

TL;DR

This work establishes a tight correspondence between K-polystability for normal and special test configurations for log Fano cone singularities under the delta-condition $\delta_{\mathbb T}(X,\Delta,\xi_0)\ge 1$, leveraging higher-rank finite generation and non-Archimedean Ding theory. It introduces and analyzes delta minimizers, proving they are Kollár valuations, and connects stability notions on the cone to base data on the quotient in the quasi-regular setting. The authors develop reduced uniform stability concepts via the reduced $J$-norm, showing these reduced notions are equivalent to (Ding/K-)polystability and to the corresponding reduced delta invariants. Consequently, the paper unifies various stability notions (Ding, Fut, polystability for normal and special test configurations) in the log Fano cone setting, providing a robust framework for optimal degenerations and their geometric implications. The results clarify when Ding-polystability for special configurations implies full polystability and reveal a natural reduction principle to quotient bases in the quasi-regular case.

Abstract

We prove that a log Fano cone $(X,Δ,ξ_0)$ satisfying $δ_\mathbb{T}(X,Δ,ξ_0)\ge 1$ 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 $(X,Δ,ξ_0)$ and show that it is equivalent to K-polystability.

Theorems & Definitions (97)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• proof