On the Cohomology of Discriminantal Arrangements and Orlik–Solomon Algebras

We relate the cohomology of the Orlik-Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik-Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in an arbitrary complex rank one local system. We also establish the relationship between the cohomology support loci of the complement of a discriminantal arrangement and the resonant varieties of its Orlik-Solomon algebra. Department of Mathematics Louisiana State University Baton Rouge, LA 70803 U. S. A. cohen@math.lsu.edu http://www.math.lsu.edu/~cohen 1991 Mathematics Subject Classification. Primary 52B30, 55N25; Secondary 20F36.