Fundamental group and pluridifferentials on compact Kähler manifolds

A compact Kähler manifold $X$ is shown to be simply connected if its ‘symmetric cotangent algebra’ is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective connected holomorphic map $f\colon X\to S$ between connected manifolds induces an isomorphism of fundamental groups if its smooth fibres are as above, and if $X$ is Kähler.