The GKZ hypergeometric $\mathcal D$-module

TL;DR

The paper defines and analyzes the GKZ hypergeometric $\mathcal{D}$-module $\mathrm{Hyp}_{\overline{\boldsymbol\gamma}}$ via cohomological and Fourier-transform techniques, establishing holonomicity and a rank formula independent of $\boldsymbol\gamma$ on a nondegenerate locus. It connects the GKZ module to a concrete de Rham model using differential forms with logarithmic poles on toric compactifications, and provides explicit complexes computing $\mathrm{Hyp}_{\overline{\boldsymbol\gamma}}$. The main contributions include showing $\mathrm{Hyp}_{\overline{\boldsymbol\gamma}}$ is holonomic and, over a Zariski open set parametrizing nondegenerate Laurent polynomials, yields an integrable connection of rank $n!\mathrm{vol}(\Delta_\infty)$, with a robust toric-geometry framework underpinning the construction. The results unify the A-hypergeometric system with a GKZ D-module realization, suggesting a broad motive-theoretic perspective across de Rham, $\ell$-adic, and $p$-adic realizations.

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 GKZ hypergeometric $\mathcal D$-module using cohomological functors, which is closely related to the $A$-hypergeometric $\mathcal D$-module. We prove the GKZ hypergeometric $\mathcal D$-module is holonomic and is an integrable connection of rank $n!\mathrm{vol}(Δ_\infty)$ on the Zariski open subset parametrizing nondegenerate Laurent polynomials, where $Δ_\infty$ is the Newton polytope at $\infty$.

Theorems & Definitions (22)

• Remark 3
• Definition 4
• Theorem 5
• Remark 6
• Proposition 8
• Remark 9
• Proposition 1.1
• proof
• Proposition 2.2
• Lemma 2.3