# $A$-hypergeometric systems and relative cohomology

## TL;DR

This paper links the solutions of $A$-hypergeometric $\,D$-modules to relative cohomology of toric varieties, extending previous results and establishing the existence of rank one points in Calabi--Yau intersections.

## Contribution

It demonstrates that the solution space of certain $A$-hypergeometric systems can be described via relative cohomology, generalizing earlier work and proving new existence results.

## Key findings

- Solution space identified with relative cohomology groups
- Generalization of previous results by Huang, Lian, Yau, and Zhu
- Existence of rank one points for Calabi--Yau complete intersections

## Abstract

We investigate the space of solutions to certain $A$-hypergeometric $\mathscr{D}$-modules, which were defined and studied by Gelfand, Kapranov, and Zelevinsky. We show that the solution space can be identified with certain relative cohomology group of the toric variety determined by $A$, which generalizes the results of Huang, Lian, Yau, and Zhu. As a corollary, we also prove the existence of rank one points for Calabi--Yau complete intersections in toric varieties.