On abelian principal bundles

摘要

Let T be a complex torus and E T a holomorphic principal T-bundle over a connected complex manifold M. We prove that the total space of E T admits a K?hler structure if and only if M admits a K?hler structure and E T admits a flat holomorphic connection whose monodromy preserves a K?hler form on T. If E T admits a K?hler structure, then $ H* (E_T ,user2{mathbb{C}}) $ is isomorphic to $ H* (M,user2{mathbb{C})} otimes _{user2{mathbb{C}}} H* (T,user2{mathbb{C}}) $ .