Calabi problem for manifolds with edge-cone singularities

TL;DR

This paper introduces a novel method for solving the Calabi problem on complex manifolds with edge-cone singularities, extending classical techniques to singular settings with multiple hypersurfaces.

Contribution

It develops a new approach using good reference metrics to adapt Aubin-Yau estimates for manifolds with edge-cone singularities, including multiple hypersurfaces.

Findings

Extension of classical a priori estimates to edge-cone singularities

Method applicable to multiple hypersurfaces with normal crossings

Generalization of smooth case techniques to singular manifolds

Abstract

In this note, we propose a new approach to solving the Calabi problem on manifolds with edge-cone singularities of prescribed angles along complex hypersurfaces. It is shown how the classical approach of Aubin-Yau in derving {\it a priori} estimates for the complex hessian can be made to work via adopting a \emph{good reference metric} and studying equivalent equations with different referrence metrics. This further allows extending much of the methods used in the smooth setting to the edge setting. These results generalise to the case of multiple hypersufaces with possibly normal crossing.