K\"ahler metrics with cone singularities along a divisor of bounded Ricci curvature

TL;DR

This paper proves the existence of K"ahler metrics with cone singularities and bounded Ricci curvature along a divisor, and offers an alternative proof for the existence of K"ahler-Einstein metrics with cone singularities.

Contribution

It establishes the existence of cone singularity K"ahler metrics with bounded Ricci curvature in any positive cohomology class and provides a new proof for K"ahler-Einstein metrics with cone singularities.

Findings

Existence of cone singularity K"ahler metrics with bounded Ricci curvature

Alternative proof for K"ahler-Einstein metrics with cone singularities

Construction of metrics in any positive cohomology class

Abstract

Let $D$ be a smooth divisor in a compact complex manifold $X$ and let $\beta \in (0,1)$. We show that in any positive co-homology class on $X$ there is a K\"ahler metric with cone angle $2\pi\beta$ along $D$ which has bounded Ricci curvature. We use this result together with the Aubin-Yau continuity method to give an alternative proof of a well-known existence theorem for Kahler-Einstein metrics with cone singularities.