On the Gauss-Manin connection and hypergeometric representations of the pure braid group
by
Herbert Kanarek
Instituto de Matematicas, UNAM

We study local systems arising from flat line bundles over topologically trivial families U --> S of hyperplane complements in Pⁿ. Imposing some genericity condition on the monodromy, one knows that fiberwise the cohomology of 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 hyperplanes and moving one additional hyperplane. The line bundle is the structure sheaf, endowed with the connection d_rel+\omega, for a logarithmic relative differential form \omega. In this situation we construct the Gauß-Manin connection Ñ on Rⁿ\pi_* (\Omega^\bullet _X/S(logD'), d_rel+\omega). We show that these sheaves are free. Using the combinatorics of A' we give a basis for these sheaves and an algorithm to express the connection Ñ in this basis.
These results can be seen as a generalization of the hypergeometric functions.
We apply this method for the case when A is a configuration of n+2 hyperplanes in general position in Pⁿ. As discriminant of A we have the braid configuration. Calculating the monodromy of this system we obtain hypergeometric representations of the pure braid group.
We illustrate the method with some examples.

Date received: May 24, 1999