  We give a lower bound on the Hodge number h^{2,0}(X), where X is an irregular
compact K\"ahler (or smooth complex projective) variety, in terms of the
minimal rank of an element in the kernel of the wedge product map \psi_2:
\Lambda^2 H^0(X,\Omega_X^1) -> H^0(X,\Omega_X^2). As a consequence, we obtain a
generalization to higher dimensions of the Castelnuovo-de Franchis inequality
for surfaces, improving some results of Lazarsfeld and Popa and Lombardi for
threefolds and fourfolds.
