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Représentations linéaires de groupes kählériens et de leurs analogues projectifs

In this note we establish the following result (announced in a previous work): if a linear group is the image of a representation of a Kähler group, then it has a finite index subgroup which is the image of a representation of the fundamental group of a smooth projective variety. In particular, a Kähler group which is linear is virtually projective.