A-hypergeometric modules and Gauss–Manin systems.

Abstract Let A be a d × n integer matrix. Gel′fand et al. 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 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. 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 C [ 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. [ABSTRACT FROM AUTHOR]