On uniform log $K$-stability for constant scalar curvature Kähler cone metrics

TL;DR

This work advances the log Yau–Tian–Donaldson program for constant scalar curvature Kähler (cscK) cone metrics by showing that the existence of a cscK cone metric along a divisor forces log $K$-polystability and $G$-uniform log $K$-stability. It develops a comprehensive variational framework in the log setting, introducing and relating the log $K$-energy, log entropy, $D$-functional, and $J$-functional, and connects these to non-Archimedean stability via Boucksom–Hisamoto–Jonsson theory. The paper proves existence results for ample divisors of large degree, establishes openness and stability along the cscK cone path, and extends uniform log $K$-stability to certain singular varieties using blow-up formulas for the log Donaldson–Futaki invariant. It also provides precise alpha-invariant and entropy-threshold criteria that guarantee coercivity of the log $K$-energy, yielding concrete geometric consequences and a rich set of algebraic and analytic tools for studying cone metrics and stability in both smooth and singular settings.

Abstract

We prove that the existence of constant scalar curvature Kähler metrics with cone singularities along a divisor implies log $K$-polystability and $G$-uniform log $K$-stability, where $G$ is the automorphism group which preserves the divisor. We also show that a constant scalar curvature Kähler cone metric along an ample divisor of sufficiently large degree always exists. We further show several properties of the path of constant scalar curvature Kähler cone metrics and discuss uniform log $K$-stability of normal varieties.

Theorems & Definitions (161)

• Conjecture 1.1: Log YTD conjecture
• Theorem 1.2: Theorem \ref{['log K']}
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7: Theorem \ref{['log K']}
• Conjecture 1.8: Log YTD conjecture for uniform stability
• Theorem 1.9: Theorem \ref{['log K']}
• Theorem 1.10: Theorem \ref{['thmalphainv']}