Abelian connectedness of compact Kahler manifolds

Let X be a compact Kahler manifold. It is shown in particular that pi(1) (X) is virtually abelian (resp. nilpotent, polycyclic) if any two of its points can be joined by a chain of irreducible compact analytic subsets Z(i) with smooth model <(Z)over tilde (i)> such that pi(1) (<(Z)over tilde (i)>) is virtually abelian (resp. nilpotent, polycyclic).