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

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.