A generalization of the Castelnuovo-de Franchis inequality

We give a lower bound on the Hodge number h^{2,0}(X), where X is an irregular compact Kähler (or smooth complex projective) variety, in terms of the minimal rank of an element in the kernel of the wedge product map ψ_2: Λ^2 H^0(X,Ω_X^1) -> H^0(X,Ω_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.

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