Degeneration of Kahler-Einstein manifolds I: The normal crossing case

Abstract

In this paper we prove that the Kahler-Einstein metrics for a degeneration family of Kahler manifolds with ample canonical bundles converge in the sense of Cheeger-Gromov to the complete Kahler-Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil-Peterson metric in this case.