Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality

We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis inequality for surfaces. The proof is based on the theory of Generic Vanishing and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative algebra. Along the way we give a positive answer, in the setting of K"ahler manifolds, to a question of Green-Lazarsfeld on the vanishing of higher direct images of Poincar'e bundles. We indicate generalizations to arbitrary Fourier-Mukai transforms.