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Transactions of the American Mathematical Society

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Cohomology of the complement of a free divisor


Authors: Francisco J. Castro-Jiménez, Luis Narváez-Macarro and David Mond
Journal: Trans. Amer. Math. Soc. 348 (1996), 3037-3049
MSC (1991): Primary 32S20, 32S25, 14F40; Secondary 52B30, 58C27
DOI: https://doi.org/10.1090/S0002-9947-96-01690-X
MathSciNet review: 1363009
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Abstract | References | Similar Articles | Additional Information

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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  • 1. E. Brieskorn, Singular elements of semi-simple algebraic groups, in Actes Congres Intern. Math. 1970, vol. 2, 279-284. MR 55:10720
  • 2. E.Brieskorn, Sur le groupe de tresses (d'apres V.I. Arnol'd), Sem. Bourbaki 1971/72, Lecture Notes in Math. 317, Springer Verlag, Berlin, 1973, 21-44. MR 54:10660
  • 3. R. Ephraim, Isosingular loci and the cartesian product stucture of complex analytic singularities, Trans. Amer. Math. Soc. 241 (1978), 357-371. MR 80i:32027
  • 4. A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publ.Math. de l'I.H.E.S. 29 (1966) 95-103. MR 33:7343
  • 5. A. Grothendieck, Local Cohomology, Lecture Notes in Math. 41, Springer Verlag, Berlin, 1967. MR 37:219
  • 6. E.J.N. Looijenga, Isolated Singular Points on Complete Intersections, London Math. Soc. Lecture Note Ser. 77, 1984. MR 86a:32021
  • 7. J.N. Mather, Stability of $C^{\infty }$ mappings IV: Classification of stable germs by $\mathbf {R} $-algebras, Publ. Math. I.H.E.S. 37, 1970, 223-248. MR 43:1215b
  • 8. J.N. Mather, Stability of $C^{\infty }$ mappings V: Transversality, Advances in Math. 4 (1970), 301-336. MR 43:1215c
  • 9. J.N. Mather, Stability of $C^{\infty }$ mappings VI: The nice dimensions, Proceedings of the Liverpool Singularities Symposium Vol.1, Lecture Notes in Math. 192, Springer, Berlin, 1971, 205-255. MR 45:2727
  • 10. D. Northcott, Injective envelopes and inverse polynomials, J. London Math. Soc. 8, 1974, 290-296. MR 50:13003
  • 11. P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren der mathematischen Wissenschaften 300, Springer Verlag, Berlin, etc., 1992. MR 94e:52014
  • 12. K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sec. 1 A, 27 (1980), 266-291. MR 83h:32023
  • 13. C.T.C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc., 13 (1981), 481-539. MR 83i:58020

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Additional Information

Francisco J. Castro-Jiménez
Affiliation: Departamento de Álgebra, Computación, Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41012 Sevilla, Spain
Email: castro@atlas.us.es

Luis Narváez-Macarro
Affiliation: Departamento de Álgebra, Computación, Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41012 Sevilla, Spain
Email: narvaez@atlas.us.es

David Mond
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, England
Email: mond@maths.warwick.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-96-01690-X
Received by editor(s): November 4, 1994
Additional Notes: The first two authors were supported by DGICYT PB94-1435.
Article copyright: © Copyright 1996 American Mathematical Society

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