On the exterior power structure of the cohomology groups for the general hypergeometric integral

TL;DR

This paper investigates the algebraic de Rham cohomology groups linked to Gelfand hypergeometric functions, revealing their structure as exterior powers of simpler cohomology groups in special cases.

Contribution

It demonstrates that the cohomology groups for certain hypergeometric integrals can be expressed as exterior powers of one-dimensional cases, providing new structural insights.

Findings

Cohomology groups are expressible as exterior powers in special subsets.

The structure simplifies the understanding of hypergeometric integrals.

Results apply to both generic and confluent hypergeometric functions.

Abstract

In this article we study the exterior power structure of the algebraic de Rham cohomology group associated with the Gelfand hypergeometric function and its confluent family. The hypergeometric function F(z) is a function on the Zariski open subset $Z_{r+1}\subset\mat{r+1,N}$, called the generic stratum, defined by an r-dimesional integral on $\Ps^{r}$. For $z\in Z_{r+1}$, the algebraic de Rham cohomology group is associated to the integral. When z belongs to the particular subset of $Z_{r+1}$, called the Veronese image, we show that this cohomology group can be expressed as the exterior power product of the de Rham cohomology group associated with the hypergeometric function defined by 1-dimensional integral.