Cohomology of the complement of a free divisor

Abstract

We prove that if D D is a “strongly quasihomogeneous" free divisor in the Stein manifold X X , and U U is its complement, then the de Rham cohomology of U U can be computed as the cohomology of the complex of meromorphic differential forms on X X with logarithmic poles along D D , with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather’s nice dimensions (and in particular the discriminants of Coxeter groups).