A generalization of the Castelnuovo-de Franchis inequality (Q2807096)

For an irregular compact complex surface \(S\), the classical Castelnuovo-de Franchis theorem states that \(S\) admits a fibration over a curve of genus \(\geq 2\) if and only if there are two distinct non-zero holomorphic \(1\)-forms \(\omega_1\), \(\omega_2\) on \(S\) such that \(\omega_1 \wedge \omega_2 = 0\). A sufficient criterion for the existence of such \(\omega_1\), \(\omega_2\) is the \textit{Castelnuovo-de Franchis inequality} \(p_g(S) \leq 2q(S) - 4\).NEWLINENEWLINEIn order to generalize the Castelnuovo-de Franchis inequality to higher dimensions, the article under review introduces the so-called \textit{Grassmannian BGG complexes} on an irregular compact complex Kähler manifold \(X\). Exactness properties of these complexes give rise to inequalities for the Hodge numbers \(h^{k,0}(X)\) (\(k=1,\dots,\dim X\)). The author studies the case \(k=2\) in detail and obtains a lower bound for \(h^{2,0}(X)\) in terms of the minimal rank of an element in the kernel of the natural map \(\bigwedge^2 H^0(\Omega^1_X) \to H^0(\Omega^2_X)\).NEWLINENEWLINEUsing a generalized version of the Castelnuovo-de Franchis theorem proved by \textit{F. Catanese} [Invent. Math. 104, No.~2, 263--289; Appendix 289 (1991; Zbl 0743.32025)] and \textit{Z. Ran} [Invent. Math. 62, 459--479 (1980; Zbl 0474.14016)], the article obtains a lower bound for \(h^{2,0}(X)\) in terms of \(q(X)\) and \(\dim X\) in the case that \(X\) does not carry a \textit{higher irrational pencil} (i.e., a certain fibration analogous to a surface fibered over a curve of genus \(\geq 2\)), thus improving certain inequalities proved earlier by various authors (see the reviewed article for references).