A conical approximation of constant scalar curvature K\"{a}hler metrics of Poincar\'{e} type

TL;DR

This paper demonstrates that constant scalar curvature Kähler metrics of Poincaré type on a polarized manifold minus a hypersurface can be approximated by cone singularity metrics, linking to log K-semistability.

Contribution

It establishes a new approximation method for Poincaré type metrics using cone singularities under specific automorphism conditions.

Findings

Approximation of Poincaré type metrics by cone singularity metrics.

Implication of log K-semistability at zero angle.

Conditions for the existence of such approximations.

Abstract

Let $(X,L_X)$ be a polarized manifold and $D$ be a smooth hypersurface such that $D \in | L_X |$. In this paper, we show that if there is no nontrivial holomorphic vector field on $D$ and ${\rm Aut}_0 ((X,L_X); D)$ is trivial, then constant scalar curvature K\"{a}hler metrics of Poincar\'{e} type on $X \setminus D$ can be approximated by constant scalar curvature K\"{a}hler metrics with cone singularities of sufficiently small angle along $D$. This result implies log K-semistability of $((X,L_X);D)$ with angle 0.