Holomorphic maps onto Kahler manifolds with non-negative Kodaira dimension

This paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space X to a compact Kahler manifold Y. We will show that when the target has non-negative Kodaira dimension, all deformations of surjective holomorphic maps X Y come from automorphisms of an unramified covering of Y and the underlying reduced varieties of associated components of Hol(X, Y) are complex tori. Under the additional assumption that Y is projective algebraic, this was proved in [7]. The proof in [7] uses the algebraicity in an essential way and cannot be generalized directly to the Kahler setting. A new ingredient here is a careful study of the infinitesimal deformation of orbits of an action of a complex torus. This study, combined with the result for the algebraic case, gives the proof for the Kahler setting.