Wiens, Jonathan; Yuzvinsky, Sergey De Rham cohomology of logarithmic forms on arrangements of hyperplanes. (English) Zbl 0948.52014 Trans. Am. Math. Soc. 349, No. 4, 1653-1662 (1997). Summary: 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. Cited in 6 Documents MSC: 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 14F40 de Rham cohomology and algebraic geometry 05B35 Combinatorial aspects of matroids and geometric lattices Keywords:de Rham cohomology; complex of logarithmic differential forms; hyperplane arrangements; Orlik-Solomon algebra PDFBibTeX XMLCite \textit{J. Wiens} and \textit{S. Yuzvinsky}, Trans. Am. Math. Soc. 349, No. 4, 1653--1662 (1997; Zbl 0948.52014) Full Text: DOI References: [1] Egbert Brieskorn, Sur les groupes de tresses [d’après V. I. Arnol\(^{\prime}\)d], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Springer, Berlin, 1973, pp. 21 – 44. Lecture Notes in Math., Vol. 317 (French). [2] Castro, F., Narváez, L., and Mond, D.: Cohomology of the complement of a free divisor, preprint, 1994. · Zbl 0862.32021 [3] Roger Godement, Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). · Zbl 0080.16201 [4] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95 – 103. · Zbl 0145.17602 [5] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001 [6] Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167 – 189. · Zbl 0432.14016 [7] Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. · Zbl 0757.55001 [8] Orlik, P., Terao, H.: Arrangements and Milnor fibers, preprint, 1993. · Zbl 0813.32033 [9] Lauren L. Rose and Hiroaki Terao, A free resolution of the module of logarithmic forms of a generic arrangement, J. Algebra 136 (1991), no. 2, 376 – 400. · Zbl 0732.13010 [10] L. Solomon and H. Terao, A formula for the characteristic polynomial of an arrangement, Adv. in Math. 64 (1987), no. 3, 305 – 325. · Zbl 0625.05001 [11] Hiroaki Terao, Forms with logarithmic pole and the filtration by the order of the pole, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 673 – 685. · Zbl 0429.32015 [12] Jürgen Stückrad and Wolfgang Vogel, Buchsbaum rings and applications, Springer-Verlag, Berlin, 1986. An interaction between algebra, geometry and topology. J. Stückrad and W. Vogel, Buchsbaum rings and applications, Mathematische Monographien [Mathematical Monographs], vol. 21, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986. An interaction between algebra, geometry, and topology. · Zbl 0606.13018 [13] Terao, H., Yuzvinsky, S.: Logarithmic forms on affine arrangements, preprint, 1994. · Zbl 0848.57022 [14] Sergey Yuzvinsky, Cohomology of local sheaves on arrangement lattices, Proc. Amer. Math. Soc. 112 (1991), no. 4, 1207 – 1217. · Zbl 0758.32014 [15] Sergey Yuzvinsky, On generators of the module of logarithmic 1-forms with poles along an arrangement, J. Algebraic Combin. 4 (1995), no. 3, 253 – 269. · Zbl 0826.52017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.