Fibred Kähler and quasi-projective groups

Abstract

We formulate a new theorem giving several necessary and sufficient conditions in order that a surjection of the fundamental group πι^) of a compact Kahler manifold onto the fundamental group Π^ of a compact Riemann surface of genus g ^ 2 be induced by a holomorphic map. For instance, it suffices 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 —> U g —> 1, where H is finitely generated, to be the fundamental group of a compact Kahler 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 Kahler manifolds) we first extend the previous result to the non-compact case. We are finally able to give a topo- logical 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.