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\(A\)-hypergeometric modules and Gauss-Manin systems. (English) Zbl 1408.14032
Author’s abstract: Let \(A\) be a \(d \times n\) integer matrix. Gel’fand et al. [4] proved that most \(A\)-hypergeometric systems have an interpretation as a Fourier-Laplace transform of a direct image. The set of parameters for which this happens was later identified by Schulze and Walther [21] as the set of not strongly resonant parameters of \(A\). A similar statement relating \(A\)-hypergeometric systems to exceptional direct images was proved by Reichelt [16]. In this article, we consider a hybrid approach involving neighborhoods \(U\) of the torus of \(A\) and consider compositions of direct and exceptional direct images. Our main results characterize for which parameters the associated \(A\)-hypergeometric system is the inverse Fourier-Laplace transform of such a “mixed Gauss-Manin” system.
In order to describe which \(U\) work for such a parameter, we introduce the notions of fiber support and cofiber support of a \(D\)-module.
If the semigroup ring \(\mathbb{C} [\mathbb{N} A]\) is normal, we show that every \(A\)-hypergeometric system is “mixed Gauss-Manin”. We also give an explicit description of the neighborhoods \(U\) which work for each parameter in terms of primitive integral support functions.

MSC:
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14D06 Fibrations, degenerations in algebraic geometry
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
Software:
Macaulay2
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References:
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