A conical approximation of constant scalar curvature Kähler metrics of Poincaré type

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 ${\mathrm{Aut}}_0((X,L_X);D)$ is trivial, then constant scalar curvature Kähler metrics of Poincaré type on $X\setminus D$ can be approximated by constant scalar curvature Kähler metrics with cone singularities of sufficiently small angle along $D$. This result implies log K-semistability of $((X,L_X);D)$ with angle 0.

18 pages, no figures