β-nbc bases for cohomology of local systems on hyperplane complements

1994/12/12 by Michael Falk, Falk, Michael, Hiroaki Terao +1
Mathematics · #52B30 #Algebraic Geometry (math.AG) #FOS: Mathematics #alg-geom #math.AG #msc:52B30

paper · pdf · doi:10.48550/arxiv.alg-geom/9412009

minor additions to introduction and references, 14 pages. AMSLaTeX v1.2

arxiv created 1995/05/24 · arxiv updated 2009/11/30

Abstract

We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes \A. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement \A. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of \A. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.

Related

• Rank one local systems on complements of hyperplanes and Aomoto complexes
• Minimality of hyperplane arrangements and basis of local system cohomology
• Cohomology of the complement to an elliptic arrangement
• Monodromy of hypergeometric functions arising from arrangements of hyperplanes
• Non-vanishing of the twisted cohomology on the complement of hypersurfaces