Let be a complex hyperbolic manifold of divergence type and M a complex manifold which may not be compact nor complex hyperbolic. In this talk, we shall consider the (strong) rigidity for non-constant holomorphic maps of N to M. Generally, the rigidity does not hold. Hence, we shall consider sufficient conditions for the manifold M to have the rigidity. As an application, we shall establish a finiteness theorem for holomorphic maps of N to M if is finitely generated and if M is compact.
Also, we shall construct examples of the manifold M with the rigidity for holomorphic mappings of N and consider related topics.