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We extend to compact K\"ahler manifolds some classical results on linearrepresentation of fundamental groups of complex projective manifolds. Ourapproach based on an interversion lemma for fibrations with tori versus generaltype manifolds as fibers gives a refinement of the classical work of Zuo. Weextend to the kahler case some general results on holomorphic convexity ofcoverings such as the linear shafarevich conjecture. In the first version, theproof of the statement that a linear Kahler group is virtuallycomplex-projective was wrong. We removed it from this new version. The proofwill be given in a forthcoming work.

Repr\'esentations lin\'eaires des groupes k\"ahl\'eriens : Factorisations et conjecture de Shafarevich lin\'eaire