Degeneration of Kähler-Einstein Manifolds II: The Toroidal Case

Abstract

In this paper we prove that the Kähler-Einstein metrics for a toroidal canonical degeneration family of Kähler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete Kähler-Einstein metric on the smooth part of the central fiber when the base locus of the degeneration family is empty. We also prove the incompleteness of the Weil-Peterson metric in this case.

The assumption of simple in the toroidal degeneration is removed using base extension