Monodromy of hypergeometric functions arising from arrangements of hyperplanes

Abstract

Given an arrangement of hyperplanes in $¶^n$, 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 Gauß-Manin connection for this case. We give a basis for the Gauß-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.