Laurent Polynomials, GKZ-hypergeometric Systems and Mixed Hodge Modules

TL;DR

This paper establishes a connection between Laurent polynomial families, GKZ systems, and mixed Hodge modules, revealing structural properties and monodromy behavior of GKZ systems.

Contribution

It constructs a morphism linking Gauss-Manin systems and GKZ systems, enabling the application of mixed Hodge module theory to GKZ systems and analyzing their monodromy.

Findings

Kernel and cokernel are simple free O-modules

GKZ systems can be endowed with mixed Hodge module structures

GKZ systems exhibit quasi-unipotent monodromy

Abstract

Given a family of Laurent polynomials, we will construct a morphism between its (proper) Gauss-Manin system and a direct sum of associated GKZ systems. The kernel and cokernel of this morphism are very simple and consist of free O-modules. The result above enables us to put a mixed Hodge module structure on certain classes of GKZ systems and shows that they have quasi-unipotent monodromy.