Calabi problem for manifolds with edge-cone singularities

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.