Degeneration of Kähler-Einstein Manifolds II: The Toroidal Case

This paper is a sequel of [8]. In algebraic geometry, when discussing the compactification of the moduli space of complex manifold X with ample canonical bundle KX , it is necessary to consider holomorphic degeneration family π : X → B, where Xt = π(t) are smooth for t 6= 0, X and X0 are QGorenstein, such that the canonical bundle of Xt for t 6= 0 and the dualizing sheaf of X0 are ample. We will call such degeneration canonical degeneration. We are interested in studying the degeneration behavior of the family of Kähler-Einstein metrics gt on Xt when t approaches 0. Following his seminal proof of Calabi conjecture ([13]), Yau ([11]) initiated the program of studying the application of Kähler-Einstein metrics to algebraic geometry with the belief that the behavior of Kähler-Einstein metrics should reflect the topological, geometric and algebraic structure of the underlying complex algebraic manifolds. According to this philosophy, one would expect the metric degeneration of the Kähler-Einstein manifolds to be closely related to the algebraic degeneration of the underlying algebraic manifolds. In [9], Tian made the first important contribution along this direction. He proved that the Kähler-Einstein metrics on