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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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  • 1. Brieskorn, E.: Sur les groupes de tresses, In: Séminaire Bourbaki 1971/1972, Lecture Notes in Math. 317 (1973), 21-44. MR 54:10660
  • 2. Castro, F., Narváez, L., and Mond, D.: Cohomology of the complement of a free divisor, preprint, 1994.
  • 3. Godement, R.: Topologie algébrique et théorie des faisceaux. Hermann, Paris, 1958. MR 21:1583
  • 4. Grothendieck, A.: On the de Rham cohomology of algebraic varieties, Publ. Math. de l'I.H.E.S. 29 (1966), 95-103. MR 33:7343
  • 5. Hartshorne, R.: Algebraic Geometry. Springer Verlag, 1977. MR 57:3116
  • 6. Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167-189. MR 81e:32015
  • 7. Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren der Math. Wiss. 300, Springer Verlag, 1992. MR 94e:52014
  • 8. Orlik, P., Terao, H.: Arrangements and Milnor fibers, preprint, 1993.
  • 9. Rose, L., Terao, H.: A free resolution of the module of logarithmic forms of a generic arrangement, J. of Algebra 136 (1991), 376-400. MR 93h:32048
  • 10. Solomon, L., Terao, H.,: A formula for the characteristic polynomial of an arrangement, Adv. Math. 64 (1987), 305-325. MR 88m:32022
  • 11. Terao, H.,: Forms with logarithmic pole and the filtration by the order of the pole, Proc. Internat. Sympos. on Algebraic Geometry, Kyoto, 1977, Kinokuniya, Tokyo, 1978, 673-685. MR 82b:14004
  • 12. Stückrad, J., Vogel, W.: Buchsbaum Rings and Applications, Springer-Verlag, 1986. MR 88h:13011b
  • 13. Terao, H., Yuzvinsky, S.: Logarithmic forms on affine arrangements, preprint, 1994.
  • 14. Yuzvinsky, S.: Cohomology of local sheaves on arrangement lattices, Proceedings of AMS 112 (1991), 1207-1217. MR 91j:52016
  • 15. Yuzvinsky, S.: Generators of the module of logarithmic 1-forms with poles along an arrangement, J. Algebra Combinatorics 4 (1995), 253-269. MR 96b:32044

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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

Received by editor(s): November 15, 1994
Article copyright: © Copyright 1997 American Mathematical Society

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