On the better behaved version of the GKZ hypergeometric system

TL;DR

This paper introduces a modified GKZ hypergeometric system tailored for finitely generated abelian groups, ensuring the system's rank matches expectations and exploring its connections to Hodge structures in algebraic geometry.

Contribution

It presents a better behaved version of the GKZ hypergeometric system with consistent rank properties and provides comprehensive proofs of its fundamental features.

Findings

The new system has the expected rank in all cases.

Connections established between the system and Hodge structures.

Provides self-contained proofs of key properties.

Abstract

We consider a version of the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinski (GKZ) suited for the case when the underlying lattice is replaced by a finitely generated abelian group. In contrast to the usual GKZ hypergeometric system, the rank of the better behaved GKZ hypergeometric system is always the expected one. We give largely self-contained proofs of many properties of this system. The discussion is intimately related to the study of the variations of Hodge structures of hypersurfaces in algebraic tori.