We extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach based on an interversion lemma for fibrations with tori versus general type manifolds as fibers gives a refinement of the classical work of Zuo. We extend to the kahler case some general results on holomorphic convexity of coverings such as the linear shafarevich conjecture. In the first version, the proof of the statement that a linear Kahler group is virtually complex-projective was wrong. We removed it from this new version. The proof will be given in a forthcoming work.