Weak positivity for Hodge modules (Q727712)

This paper deals with the study of positivity properties of extensions of variations of Hodge structure. The construction starts with a smooth projective complex variety \(X\) and an open dense subset \(U\). When \(X=U\), Griffiths proved that the lowest term of the Hodge filtration is a nef vector bundle. Fujita and later Kawamata extended this result to the case of a simple normal crossing divisor, together with monodromy properties. The authors here present a further generalisation in the setting of Hodge modules. In particular they first recall the notion of \textit{weak positivity}, introduced by \textit{E. Viehweg} in [Adv. Stud. Pure Math. 1, 329--353 (1983; Zbl 0513.14019)]. Weak positivity is investigated in the case of \(M\), the (unique) pure Hodge module extension on \(X\) coming from a polarizable VHS on \(U\) with unipotent local monodromies. To \(M\), for any integer \(p\), it is associated a set of natural Kodaira-Spencer type homomorphisms \(\theta_p: \mathrm{gr}^F_p \mathcal{M} \to \mathrm{gr}^F_{p+1} \mathcal{M} \otimes \Omega^1_X\). Denote by \(K_p(M)\) the kernel of the \(p\)-th homomorphism above. Then the authors prove how the duals of the sheaves \(K_p(M)\) are weakly positive. This is useful when working in families: indeed another work of the first author is quoted, where the above results are useful in the setting of Viehweg's hyperbolicity conjecture.