  We study irrational pencils with isolated critical points on compact
aspherical complex manifolds. We prove that if the set of critical points is
nonempty, the homology of the kernel of the morphism induced by the pencil on
fundamental groups is not finitely generated. This generalizes a result by
Dimca, Papadima and Suciu. By considering self-products of the
Cartwright-Steger surface, this allows us to build new examples of smooth
projective varieties whose fundamental group has a non-finitely generated
homology.
