Some criteria for uniform K-stability

TL;DR

The paper shows that uniform K-stability of log Fano pairs is equivalent to a positive lower bound on the β-invariant, formalized as $β_{X,Δ}(E) \ge ε\, j_{X,Δ}(E)$ for all divisors. It provides a boundary-perturbation criterion and a β-based criterion that together characterize uniform K-stability, and introduces the uniformity measure $u(X,Δ)=\inf_E β/j$ with $u>0$ signaling uniform stability (and $δ(X,Δ)=1$ signaling the boundary case $u=0$). The work extends to twisted K-stability, showing maximal twist behavior when $δ(X)\le 1$, and proves that for $0<μ<δ(X)$ the twisted theory yields uniform stability, while at $μ=δ(X)$ one obtains semistability without uniformity. It further connects these stability notions to the Optimal Destabilization Conjecture, establishing equivalences with the existence of divisorial $δ$-minimizers and the corresponding δ-twisted zero Futaki invariant, thereby linking divisorial valuations, test configurations, and lc/complement techniques in a unified framework.

Abstract

We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to $β$-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stability in the twisted setting.

Theorems & Definitions (43)

• Theorem 1.1: =Theorem \ref{['uks via boundary']}
• Theorem 1.2
• Theorem 1.3: =Theorem \ref{['maximal twist']}
• Theorem 1.4: =Theorem \ref{['equivalence']}
• Definition 2.1: Fuj19
• Definition 2.2
• Lemma 2.3
• Remark 2.4
• Definition 2.5: Fuj19
• Remark 2.6