On the convergence and collapsing of Kahler metrics

Abstract

In this paper we consider the convergence and collapsing of Kahler manifolds. While the convergence and collapsing of Riemannian manifolds have been discussed by many people and applied to many fields, how to generalize it to Kahler case is not apriorily clear. Our paper is an attempt in this direction. We discussed the corresponding concepts of convergence and collapsing for Kahler manifolds. We proved that when a sequence of Kahler manifolds with the fixed background complex compact manifold is not collapsing, it will converge to a complete Kahler manifold which is biholomorphic to a Zariski open set of the original background complex manifold with some possible "bubbling" on the complement of that Zariski open set. We also discussed the structure of collapsing. Especially we show the resulting Monge-Ampere foliation is holomorphic, produce some holomorphic Vector fields with respect to the foliation, and also give some applications of our results. The main methods we are using are estimates from the theories of harmonic maps and partial differential equations, some results from several complex variables, and ideas from Riemannian geometry.