Journal of the
Korean Mathematical Society JKMS

This paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space $X$ to a compact K\"ahler manifold $Y$. We will show that when the target has non-negative Kodaira dimension, all deformations of surjective holomorphic maps $X \to Y$ come from automorphisms of an unramified covering of $Y$ and the underlying reduced varieties of associated components of ${\rm Hol}(X,Y)$ 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 K\"ahler 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 K\"ahler setting.

Keywords: holomorphic maps, complex torus action

MSC numbers: 32H02, 32Q15