Local classification of chsc K\"ahler metrics with cone singularities

Abstract

Let $B\subset\mathbb{C}^n$ be a ball centred at the origin and $D=\{z^1=0\}$. Let $g$ be a K\"ahler metric of constant holomorphic sectional curvature (chsc) on $B\setminus D$, uniformly equivalent to the model cone metric of angle $2\pi\beta$, with $0<\beta<1$, and polyhomogeneous along $D$. We prove that, in suitable holomorphic coordinates $(w^1, \ldots, w^n)$ centred at the origin, the metric $g$ is the pullback of the corresponding complex space form by the map $(w^1,w^2,\dots,w^n)\longmapsto \bigl((w^1)^\beta,w^2,\dots,w^n\bigr)$.