Weight filtrations on GKZ-systems

TL;DR

This paper investigates the weight filtration of GKZ-systems with integral but non-strongly resonant parameters, revealing that the structure depends solely on the combinatorial polytopal data of the associated cone.

Contribution

It provides a combinatorial formula for the weight filtration and length of GKZ-systems, extending understanding of their mixed Hodge module structures.

Findings

The weight filtration can be computed via intersection cohomology on toric varieties.

The MHS structure can be transferred through Fourier-Laplace transforms using Fourier-Sato techniques.

The results depend only on the polytopal structure, not on the semigroup arithmetic.

Abstract

If $\beta\in\CC^d$ is integral but not a strongly resonant parameter for the homogeneous matrix $A\in\ZZ^{d\times n}$ with $\ZZ A=\ZZ^d$, then the associated GKZ-system carries a naturally defined mixed Hodge module structure. We study here in the normal case the corresponding weight filtration by computing the intersection complexes with respective multiplicities on the associated graded parts. We do this by computing the weight filtration of a Gauss-Manin system with respect to a locally closed embedding of a torus inside an affine space. We then produce a result, based on a Fourier-Sato transforms, that allows to port an MHS structure on a monodromic module through a Fourier-Laplace transform, from the Gauss-Manin system to the GKZ-system. Our results show that these data, which we express in terms of intersection cohomology groups on induced toric varieties, are purely combinatorial, and not arithmetic, in the sense that they only depend on the polytopal structure of the cone of $A$ but not on the semigroup itself. As a corollary we get a purely combinatorial formula for the length of the underlying (regular) holonomic GKZ-system, irrespective of homogeneity. In dimension up to three, and for simplicial semigroups, we give explicit generators of the weight filtration.