Compositio Mathematica

Cohomology of the Orlik–Solomon Algebras and Local Systems

Anatoly Libgober a1 and Sergey Yuzvinsky a2
a1 Department of Mathematics, University of Illinois, Chicago, IL 60607, U.S.A.
a2 Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A.

Article author query
libgober a   [Google Scholar] 
yuzvinsky s   [Google Scholar] 


The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.

Key Words: arrangements; cohomology of local systems; characteristic varieties; matrices of collections of subsets; matroids.