  We show that a conjecture of Kotschick about one-forms without zeros on
compact K\"ahler manifolds follows in the case of simple Albanese torus from a
conjecture of Bobadilla and Koll\'ar about homologically trivial fibrations. As
an application, we prove Kotschick's conjecture for compact K\"ahler manifolds
X whose first betti number is at least 2dim(X)-2 and Albanese torus is simple.
