The talk will divide into two parts. In the first part we will consider the limits of compact complex Kahler manifolds under holomorphic deformations. Let X be a smooth family of compact complex manifolds over a small disc. Suppose that all the fibers X_tare Kahler manifolds for. With some mild assumptions on metric and topology of the central fiber, we prove that the central fibe X₀is also Kahler. This part is based on the joint work with Liu Wanmin.

The second part is about the rigidity of projective manifolds with semiample canonical line bundle. This part is based on the joint work with Liu Xiao-Lei.