De Rham cohomology of logarithmic forms on arrangements of hyperplanes

Wiens, Jonathan; Yuzvinsky, Sergey

doi:10.1090/S0002-9947-97-01894-1

De Rham cohomology of logarithmic forms on arrangements of hyperplanes
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by Jonathan Wiens and Sergey Yuzvinsky PDF

Trans. Amer. Math. Soc. 349 (1997), 1653-1662 Request permission

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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