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Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modules. (English) Zbl 1315.14016
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 [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.

MSC:
14D07 Variation of Hodge structures (algebro-geometric aspects)
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
33C70 Other hypergeometric functions and integrals in several variables
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