Fibred K"ahler and quasi-projective Groups

We formulate a new theorem giving several necessary and sufficient conditions in order that a surjection of the fundamental group ${\pi }_1(X)$ of a compact K"ahler manifold onto the fundamental group ${\mathrm{\Pi }}_g$ of a compact Riemann surface of genus $g\ge 2$ 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 $1\text{ra}H\text{ra}G\text{ra}{\mathrm{\Pi }}_g\text{ra}1$, where $H$ is finitely generated, to be the fundamental group of a compact K"ahler 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 K"ahler 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.