The space of Poincar\'e type K\"ahler metrics on the complement of a divisor

TL;DR

This paper proves the uniqueness of constant scalar curvature K"ahler metrics with cusp singularities near divisors in certain compact K"ahler manifolds, extending previous constructions and theorems.

Contribution

It generalizes Chen's construction of approximate geodesics and proves an approximate Calabi-Yau theorem without requiring the ampleness of K[D].

Findings

Uniqueness of cusp singularity K"ahler metrics with constant scalar curvature when K[D] is ample.

Extension of Chen's approximate geodesic construction to this setting.

An approximate Calabi-Yau theorem applicable independently of ampleness conditions.

Abstract

Consider a divisor D with simple normal crossings in a compact K\"ahler manifold X. We show in this article that a K\"ahler metric in an arbitrary class, with constant scalar curvature and cusp singularities along the divisor is unique in this class when K[D] is ample. This we do by generalizing Chen's construction of approximate geodesics in the space of K\"ahler metrics, and proving an approximate version of the Calabi-Yau theorem, both independently of the ampleness of K[D].