Cohomology of the complement of a free divisor

Castro-Jiménez, Francisco; Narváez-Macarro, Luis; Mond, David

doi:10.1090/S0002-9947-96-01690-X

Cohomology of the complement of a free divisor
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by Francisco J. Castro-Jiménez, Luis Narváez-Macarro and David Mond

Trans. Amer. Math. Soc. 348 (1996), 3037-3049

DOI: https://doi.org/10.1090/S0002-9947-96-01690-X

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Abstract:

We prove that if $D$ is a “strongly quasihomogeneous" free divisor in the Stein manifold $X$, and $U$ is its complement, then the de Rham cohomology of $U$ can be computed as the cohomology of the complex of meromorphic differential forms on $X$ with logarithmic poles along $D$, with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather’s nice dimensions (and in particular the discriminants of Coxeter groups).

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