On abelian principal bundles

Let T be a complex torus and ET a holomorphic principal T-bundle over a connected complex manifold M. We prove that the total space of ET admits a Kähler structure if and only if M admits a Kähler structure and ET admits a flat holomorphic connection whose monodromy preserves a Kähler form on T. If ET admits a Kähler structure, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ H* (E_T ,\user2{\mathbb{C}}) $$ \end{document} is isomorphic to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ H* (M,\user2{\mathbb{C})} \otimes _{\user2{\mathbb{C}}} H* (T,\user2{\mathbb{C}}) $$ \end{document}.