Transactions of the American Mathematical Society

$\beta \,\mathbf {nbc}$ -bases for cohomology of local systems on hyperplane complements

Author(s): Michael Falk; Hiroaki Terao
Journal: Trans. Amer. Math. Soc. 349 (1997), 189-202.
MSC (1991): Primary 52B30
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Abstract: We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes ${\mathcal A}$ . 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 ${\mathcal A}$ . The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of ${\mathcal A}$ . 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.

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Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 52B30

Michael Falk
Affiliation: Department of Mathematics, Northern Arizona University, Flagstaff, Arizona 86011
Email: mjf@odin.math.nau.edu

Hiroaki Terao
Affiliation: Department of Mathematics, University of Wisconsin - Madison, Madison, Wisconsin 53704
Email: terao@math.wisc.edu

DOI: 10.1090/S0002-9947-97-01844-8
PII: S 0002-9947(97)01844-8
Received by editor(s): April 30, 1995
Additional Notes: The first author was partially supported by a Northern Arizona University Organized Research Grant
Dedicated: In memory of Michitake Kita
Copyright of article: Copyright 1997, American Mathematical Society

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