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 Kahler structure if and only if M admits a Kahler structure and E-T admits a flat holomorphic connection whose monodromy preserves a Kahler form on T. If ET admits a Kahler structure, then H*(E-T, C) is isomorphic to H*(M, C) circle times(C) H*(T, C).
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