Fibred K"ahler 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 $π_1(X)$ of a compact Kähler manifold onto the fundamental group $Π_g$ of a compact Riemann surface of genus $g \geq 2$ 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 \ra H \ra G \ra Π_g \ra 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.