  Given an arrangement of hyperplanes in $\P^n$, possibly with non-normal
crossings, we give a vanishing lemma for the cohomology of the sheaf of
$q$-forms with logarithmic poles along our arrangement. We give a basis for the
ideal $\cal J$ of relations for the Orlik-Solomon's algebra. Under certain
genericity conditions it was shown by H.~Esnault, V.~Schechtman and E.~Viehweg
that the cohomology of a local system is given by the Aomoto complex. We
generalize this result to a deformation of local systems obtained via a
deformation of our arrangement. We calculate the Gau\ss-Manin connection for
this case. We give a basis for the Gau\ss-Manin bundle for which, with help of
the basis for $\cal J$, we give then a method to calculate a representation of
this connection. From here, with the results of K-T.~Chen or P.~Deligne, one
can calculate the monodromy representation. This gives a generalization of the
hypergeometric functions.
