Locally rigid implies globally rigid in Kahler geometry

Abstract

In this paper, we study the rigidity properties of compact Kahler manifolds. Given a smooth family of compact Kahler manifolds X over the unit disk, we show that all the fibers are mutually isomorphic if the family is locally trivial at a point t_1 and the fiber X_{t_1} is non-uniruled. This proves that the locally rigid implies the global rigid in Kahler. It also can be used to prove the so called global non-deformability for non-uniruled Kahler manifolds under Kahler morphisms.