DE-RHAM COHOMOLOGY OF LOGARITHMIC FORMS ON ARRANGEMENTS OF HYPERPLANES

The paper is devoted to computation of the cohomology of the complex o f logarithmic differential forms with coefficients in rational functio ns whose poles are located on the union of several hyperplanes of a li near space over a field of characteristic zero. The main result assert s that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algebra coincides with the O rlik-Solomon algebra. Over the field of complex numbers, this means th at the cohomologies coincide with the cohomologies of the complement o f the union of the hyperplanes. We also prove that the cohomologies do not change if poles of arbitrary multiplicity are allowed on some of the hyperplanes. In particular, this gives an analogue of the algebrai c de Rham theorem for an arbitrary arrangement over an arbitrary field of zero characteristic.