The GKZ hypergeometric $\mathcal D$-module

Abstract

For an $(n\times N)$-matrix $A$ of rank $n$ with integer entries, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the $A$-hypergeometric system. We define the stable GKZ hypergeometric $\mathcal D$-module using cohomological functors, which is closely related to the $A$-hypergeometric $\mathcal D$-module and the $\mathcal D$-module underlying the better behaved GKZ system introduced by Borisov and Horja. We prove the stable GKZ hypergeometric $\mathcal D$-module is holonomic and is an integrable connection of rank $n!\mathrm{vol}(\Delta_\infty)$ on the Zariski open subset parametrizing nondegenerate Laurent polynomials, where $\Delta_\infty$ is the Newton polytope at $\infty$.