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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
Macaulay2
Full Text:
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