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 on the Evans-Griffith syzygy theorem in commutative algebra. Along the way we give a positive answer, in the setting of Kähler manifolds, to a question of Green and Lazarsfeld on the vanishing of higher direct images of Poincaré bundles. We indicate generalizations to arbitrary integral transforms.
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