$A$-Hypergeometric Modules and Gauss--Manin Systems

Let $A$ be a $d$ by $n$ 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 of $A$ 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.