Nonresonance conditions for arrangements
Annales de l'Institut Fourier, Volume 53 (2003) no. 6, p. 1883-1896
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.
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.
DOI : https://doi.org/10.5802/aif.1994
Classification:  32S22,  53C35,  55N25
Keywords: hyperplane arrangement, local system, Milnor fiber
@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, Volume 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