Strong Generic Vanishing and a Higher Dimensional Castelnuovo-de Franchis Inequality

(1) pg(X) < 2q(X)− 3 admits a fibration (i.e. a surjective morphism with connected fibres) onto a smooth curve of genus at least 2 (cf. also the modern age references [BPV], Proposition 4.2 and [Be1], C.8). This basic result in surface theory concerns in fact surfaces whose Albanese map is generically finite: otherwise the Albanese map of X automatically provides a fibration onto a smooth curve of genus at least 2, unless q(X) ≤ 1 when 2q(X)− 3 < 0. Inequality (1) can be restated as χ(ωX) < q(X)− 2. The purpose of this paper is to show that, in this form, the inequality of Castelnuovo and de Franchis has the following natural generalization in arbitrary dimension: