HOLOMORPHIC MAPS ONTO KÄHLER MANIFOLDS WITH NON-NEGATIVE KODAIRA DIMENSION

Title & Authors
HOLOMORPHIC MAPS ONTO KÄHLER MANIFOLDS WITH NON-NEGATIVE KODAIRA DIMENSION
Hwang, Jun-Muk; Peternell, Thomas;

Abstract
This paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space X to a compact $\small{K\"{a}hler}$ manifold Y. We will show that when the target has non-negative Kodaira dimension, all deformations of surjective holomorphic maps $\small{X{\rightarrow}Y}$ 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 $\small{K\"{a}hler}$ 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 $\small{K\"{a}hler}$ setting.
Keywords
holomorphic maps;complex torus action;
Language
English
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