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[NA] 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.