Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stability

TL;DR

This paper establishes the equivalence between the existence of constant scalar curvature Kähler cone metrics, properness of log K-energy, and geodesic stability, extending key conjectures from smooth to cone singularity settings and constructing new singular metrics.

Contribution

It extends the properness and stability conjectures for cscK metrics to cone singularities and introduces methods to construct and analyze singular cscK metrics with degenerations.

Findings

Equivalence of cscK cone metrics existence, properness, and stability.

Construction of singular cscK metrics with degenerations.

Proved openness and approximation properties for cscK cone metrics.

Abstract

We show that the existence of constant scalar curvature K\"ahler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjecture of the cscK problem for smooth K\"ahler metrics by Chen-Cheng to the setting of cscK cone metrics. One applications of our main results is that we introduce and construct singular cscK metrics with possible degeneration in big cohomology class. As another application, we also prove both openness and approximation property for the path of cscK cone metrics, which are paralleling to Donaldson's continuity method through K\"ahler-Einstein cone metrics in the resolution of Yau-Tian-Donaldson conjecture for Fano K\"ahler-Einstein metrics.