Some identities for hypergeometric functions have geometric meaning behind. Aomoto proposed a method to study hypergeometric functions as pairings of cycles and cocycles about 30 years ago [1]. This ingenious point of view has enabled us to yield a lot of formulas for hypergeometric functions.

The identities we have presented above
are quadratic relations for
.
We will see that it also has a geometric meaning
based on a work of Cho and Matsumoto.
They proved an analog of Riemann's period relation
for intersection numbers of cycles and cocycles and associated period
integral, of which entries are nothing but hypergeometric functions
[2].
The period relation yields a quadratic relation of hypergeometric
functions.
Therefore, the problem of deriving quadratic relations is reduced
to evaluation of intersection numbers.
Cho, Kita, Matsumoto and Yoshida gave formulas to evaluate
intersection numbers
for a class of hypergeometric functions
expressed by a definite integral
of which integrant has a normally crossing singular locus
[2], [5], [6], [7].
The GHF
has a multiple integral representation
on
but the singular locus of the integrant is not normally
crossing.
Hence, the generalized hypergeometric functions are out of the class
for general , because their method requires a construction of resolutions
of singularities and it is difficult in general.
We will introduce a different approach to study the GHF.
The GHF is expressed in terms of the single integral

The kernel function defines a locally constant sheaf of which rank is . We will evaluate intersection numbers by utilizing this integral representation. Cho, Kita, Matsumoto and Yoshida's formulas are those for locally constant sheaves associated to a product of linear forms, which are rank one sheaves. The kernel function defines a locally constant sheaf of which rank is more than . Hence, we cannot apply their formulas to the single integral representation of .

The first author studied a method to evaluate intersection numbers for cocycles with coefficients in locally constant sheaves of which rank is more than one. He applied the method for evaluating intersection numbers for cocycles associated to the Selberg type integrals [9], [10].

In this paper, we first reexamine the definition of intersection number
in view of the topological cup product,
because discussions by Kita and Yoshida [5] are not satisfactory
to apply for our problem of
and
those by the first author in [10]
are not satisfactory to be a rigorous foundation.
We will see that the method of the first author
in [9], [10]
is not only useful for computation, but also is a consequence of
a general theory of duality.
Next, we apply the method to evaluate intersection numbers
for
.
The twisted cohomology and homology groups associated to the single
integral representation of
are direct sums of *primary*
parts and *degenerate* parts.
Only the primary parts stand for
[8].
This degeneration makes the evaluation of intersection numbers
more complicated than the evaluation problem for the Selberg type integrals.