Gauss-manin connection arising from arrangements of hyperplanes

We study local systems arising from Rat line bundles over topologically tri vial families U --> S of hyperplane complements in P-n. Imposing some gener icity condition on the monodromy, one knows that fiberwise the cohomology o f the local system is concentrated in the middle dimension and is computed by the Aomoto complex, a subcomplex of global differential forms on a good compactification pi: X --> S with logarithmic poles along D' = X \ U. The families A' considered are obtained by fixing a configuration A of hype rplanes and moving one additional hyperplane. The line bundle is the struct ure sheaf, endowed with the connection d(rel) + omega, for a logarithmic re lative differential form w. In this situation we construct the GauB-Manin c onnection del on R(n)pi (*)(Omega (.)(X/S)(log D'), d(rel) + omega). We sho w that these sheaves are free. Using the combinatorics of A' we give a basi s for these sheaves and an algorithm to express the connection del in this basis. The corresponding matrix depends too much on the combinatorics of th e arrangement to be given in a closed form, but we illustrate the method wi th some examples. These results can be seen as a generalization of the hypergeometric functio ns.