  We extend to compact K\"ahler 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.
