Nonresonance conditions for arrangements  [ Conditions de non-résonance pour les arrangements ]
Annales de l'Institut Fourier, Tome 53 (2003) no. 6, p. 1883-1896
Nous démontrons un théorème d'annulation pour la cohomologie du complémentaire d'un arrangement d'hyperplans complexes à coefficients dans un système local. Ce résultat est comparé à d'autres théorèmes d'annulation et il est utilisé pour étudier les fibres de Milnor associées à des arrangements de droites et d'hypersurfaces.
We prove a vanishing theorem for the cohomology of the complement of a complex hyperplane arrangement with coefficients in a complex local system. This result is compared with other vanishing theorems, and used to study Milnor fibers of line arrangements, and hypersurface arrangements.
DOI : https://doi.org/10.5802/aif.1994
Classification:  32S22,  53C35,  55N25
Mots clés: arrangement d'hyperplans, système local, fibre de Milnor
@article{AIF_2003__53_6_1883_0,
     author = {Cohen, Daniel C. and Dimca, Alexandru and Orlik, Peter},
     title = {Nonresonance conditions for arrangements},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {6},
     year = {2003},
     pages = {1883-1896},
     doi = {10.5802/aif.1994},
     zbl = {1054.32016},
     mrnumber = {2038782},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_6_1883_0}
}
Cohen, Daniel C.; Dimca, Alexandru; Orlik, Peter. Nonresonance conditions for arrangements. Annales de l'Institut Fourier, Tome 53 (2003) no. 6, pp. 1883-1896. doi : 10.5802/aif.1994. http://www.numdam.org/item/AIF_2003__53_6_1883_0/

[1] K. Aomoto; M. Kita Hypergeometric Functions, (in Japanese), Springer-Verlag, Tokyo (1994)

[2] A. Beauville Monodromie des systèmes différentiels linéaires à pôles simples sur la sphère de Riemann (d'après A. Bolibruch), Séminaire Bourbaki, Vol. 1992/93 (Astérisque) Tome 216, Exp. No. 765, 4 (1993), pp. 103-119 | Numdam | Zbl 0796.34007

[3] A. Beilinson; J. Bernstein; P. Deligne Faisceaux Pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Soc. Math. France, Paris (Astérisque) Tome 100 (1982), pp. 5-171 | Zbl 0536.14011

[4] A. Bolibrukh The Riemann-Hilbert problem, Russian Math. Surveys, Tome 45 (1990), pp. 1-58 | Article | MR 1069347 | Zbl 0706.34005

[5] D. Cohen; A. Suciu On Milnor fibrations of arrangements, J. London Math. Soc., Tome 51 (1995), pp. 105-119 | MR 1310725 | Zbl 0814.32007

[6] J. Damon On the number of bounding cycles for nonlinear arrangements, Arrangements--Tokyo 1998, Kinokuniya, Tokyo (Adv. Stud. Pure Math) Tome 27 (2000), pp. 51-72 | Zbl 0991.32016

[7] P. Deligne Équations Différentielles à Points Singuliers Réguliers, Springer-Verlag, Berlin-New York, Lect. Notes in Math., Tome 163 (1970) | MR 417174 | Zbl 0244.14004

[8] A. Dimca Singularities and Topology of Hypersurfaces, Springer-Verlag, New York, Universitext | MR 1194180 | Zbl 0753.57001

[9] A. Dimca Sheaves in Topology (Universitext, Springer-Verlag, New York, to appear) | MR 2050072 | Zbl 1043.14003

[10] A. Dimca; J. Herzog, V. Vuletescu Eds. Hyperplane arrangements, M-tame polynomials and twisted cohomology, Commutative Algebra, Singularities and Computer Algebra, Kluwer (NATO Science Series) Tome Vol. 115 (2003), pp. 113-126 | Zbl 1046.32003

[11] A. Dimca; A. Némethi Hypersurface complements, Alexander modules and monodromy (2002) (Proceedings of the 7th Workshop on Real and Complex Singularities (Sao Carlos, 2002), to appear, preprint, math.AG/0201291) | MR 2087802 | Zbl 1067.14004

[12] A. Dimca; S. Papadima Equivariant chain complexes, twisted homology and relative minimality of arrangements (2003) (e-print, math.AG/0305266) | Numdam | MR 2060483

[13] H. Esnault; V. Schechtman; V. Viehweg Cohomology of local systems on the complement of hyperplanes, Invent. Math., Tome 109 (1992), pp. 557-561 | Article | MR 1176205 | Zbl 0788.32005

[13] H. Esnault; V. Schechtman; E. Viehweg Erratum: "Cohomology of local systems on the complement of hyperplanes", Invent. Math, Tome 112 (1993) no. 2, pp. 447 | MR 1213111 | Zbl 0794.32008

[14] H. Esnault; E. Viehweg Logarithmic de Rham complexes and vanishing theorems, Invent. Math., Tome 86 (1986), pp. 161-194 | Article | MR 853449 | Zbl 0603.32006

[15] I. M. Gelfand General theory of hypergeometric functions, Soviet Math. Dokl., Tome 33 (1986) | MR 841131 | Zbl 0037.15302

[16] M. Kashiwara; P. Schapira Sheaves on Manifolds, Springer-Verlag, Berlin, Grundlehren Math. Wiss., Tome 292 (1994) | MR 1299726 | Zbl 0709.18001

[17] T. Kohno Homology of a local system on the complement of hyperplanes, Proc. Japan Acad., Ser. A, Tome 62 (1986), pp. 144-147 | Article | MR 846350 | Zbl 0611.55005

[18] V. Kostov Regular linear systems on CP 1 and their monodromy groups, Complex analytic methods in dynamical systems (Rio de Janeiro, 1992) (Astérisque) Tome No 222 (1994), pp. 259-283 | Zbl 0814.34006

[19] A. Libgober The topology of complements to hypersurfaces and nonvanishing of a twisted de Rham cohomology, Singularities and complex geometry (Beijing, 1994), Amer. Math. Soc., Providence, RI (AMS/IP Stud. Adv. Math.) Tome 5 (1997), pp. 116-130 | Zbl 0934.14009

[20] A. Libgober Eigenvalues for the monodromy of the Milnor fibers of arrangements, Trends in Singularities, Birkhäuser (Trends Math.) (2002), pp. 141-150 | Zbl 1036.32019

[21] D. Massey Perversity, duality and arrangements in 3 , Topology Appl., Tome 73 (1996), pp. 169-179 | Article | MR 1416758 | Zbl 0867.32018

[22] P. Orlik; H. Terao Arrangements of Hyperplanes, Springer-Verlag, Berlin, Grundlehren Math. Wiss., Tome vol. 300 | MR 1217488 | Zbl 0757.55001

[23] P. Orlik; H. Terao Arrangements and Hypergeometric Integrals, Math. Soc. Japan, Tokyo, MSJ Mem., Tome 9 (2001) | MR 1814008 | Zbl 0980.32010

[24] V. Schechtman; H. Terao; A. Varchenko Local systems over complements of hyperplanes and the Kac-Kazhdan condition for singular vectors, J. Pure Appl. Algebra, Tome 100 (1995), pp. 93-102 | Article | MR 1344845 | Zbl 0849.32025

[25] A. Varchenko Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, World Scientific, River Edge, Adv. Ser. Math. Phys., Tome 21 (1995) | MR 1384760 | Zbl 0951.33001

[26] S. Yuzvinsky Cohomology of the Brieskorn-Orlik-Solomon algebras, Comm. Algebra, Tome 23 (1995), pp. 5339-5354 | Article | MR 1363606 | Zbl 0851.32027