Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modules (Q2877508)

A GKZ-hypergeometric system is a holonomic \(\mathcal D\)-module determined by an integral matrix \(A\) and a parameter vector \(\beta\). For homogeneous non-resonant systems Gelfand, Kapranov and Zelevinsky [\textit{I. M. Gelfand} et al., Adv. Math. 84, No. 2, 255--271 (1990; Zbl 0741.33011)] showed that the solution complex is isomorphic to a direct image of a local system, defined on the complement of the graph of an associated family of Laurent polynomials. In this paper for resonant but not strongly-resonant parameters a tight relation is established between certain direct sums of GKZ-systems and Gauss-Manin systems of associated families of Laurent polynomials. The results carry over to the category of mixed Hodge modules. It is shown that a homogeneous GKZ-system with non strongly-resonant, integer parameter vector \(\beta\) carries a mixed Hodge module structure. For rational \(\beta\) the GKZ-system is a direct summand in a mixed Hodge module, showing that the underlying perverse sheaf has quasi-unipotent local monodromy.