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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

De Rham cohomology of logarithmic forms
on arrangements of hyperplanes


Authors: Jonathan Wiens and Sergey Yuzvinsky
Journal: Trans. Amer. Math. Soc. 349 (1997), 1653-1662
MSC (1991): Primary 52B30, 14F40, 05B35
MathSciNet review: 1407505
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Abstract: The paper is devoted to computation of the cohomology of the complex of logarithmic differential forms with coefficients in rational functions whose poles are located on the union of several hyperplanes of a linear space over a field of characteristic zero. The main result asserts that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algebra coincides with the Orlik-Solomon algebra. Over the field of complex numbers, this means that the cohomologies coincide with the cohomologies of the complement of the union of the hyperplanes. We also prove that the cohomologies do not change if poles of arbitrary multiplicity are allowed on some of the hyperplanes. In particular, this gives an analogue of the algebraic de Rham theorem for an arbitrary arrangement over an arbitrary field of zero characteristic.


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Additional Information

Jonathan Wiens
Affiliation: Department of Mathematical Sciences, University of Alaska, Fairbanks, Alaska 99775

Sergey Yuzvinsky
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01894-1
PII: S 0002-9947(97)01894-1
Received by editor(s): November 15, 1994
Article copyright: © Copyright 1997 American Mathematical Society