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Cohomology of the Orlik–Solomon Algebras and Local Systems

Published online by Cambridge University Press:  04 December 2007

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

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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.

Keywords

arrangementscohomology of local systemscharacteristic varietiesmatrices of collections of subsetsmatroids
Type
Research Article
Information
Compositio Mathematica , Volume 121 , Issue 3 , May 2000 , pp. 337 - 361
Copyright
© 2000 Kluwer Academic Publishers