Arrangements and local systems
by
Peter Orlik
University of Wisconsin-Madison
Coauthors: Daniel C. Cohen (Louisiana State University)

We use stratified Morse theory to build a complex to compute the local system cohomology of the complement of a hyperplane arrangement.

Theorem: The linearization of this complex is the Orlik-Solomon algebra with the connection operator.

Using this result, we obtain lower bounds for the local system Betti numbers in terms of those of the Orlik-Solomon algebra, recovering a result of Libgober and Yuzvinsky. We also establish the relationship between the cohomology support loci of the complement and the resonance varieties of the Orlik-Solomon algebra for any arrangement, and show that the latter are unions of subspace arrangements in general, resolving a conjecture of Falk.

For certain local systems, our results provide new combinatorial upper bounds on the local system Betti numbers. These upper bounds enable us to prove that in nonresonant systems the cohomology is concentrated in the top dimension, without using resolution of singularities.

Date received: May 24, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadi-10.