Fibred Kähler and quasi-projective groups

We formulate a new theorem giving several necessary and su‰cient conditions in order that a surjection of the fundamental group p1ðX Þ of a compact Kähler manifold onto the fundamental group Pg of a compact Riemann surface of genus gd 2 be induced by a holomorphic map. For instance, it su‰ces that the kernel be finitely generated. We derive as a corollary a restriction for a group G, fitting into an exact sequence 1 ! H ! G ! Pg ! 1, where H is finitely generated, to be the fundamental group of a compact Kähler manifold. Thanks to the extension by Bauer and Arapura of the Castelnuovo–de Franchis theorem to the quasi-projective case (more generally, to Zariski open sets of compact Kähler manifolds) we first extend the previous result to the non-compact case. We are finally able to give a topological characterization of quasi-projective surfaces which are fibred over a (quasi-projective) curve by a proper holomorphic map of maximal rank, and we extend the previous restriction to the monodromy of any fibration onto a curve.