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Daniel C. Cohen; Alexandru Dimca; Peter Orlik
Nonresonance conditions for arrangements
(Conditions de non-résonance pour les arrangements)
Annales de l'institut Fourier, 53 no. 6 (2003), p. 1883-1896
Article PDF | Reviews MR 2038782 | Zbl 1054.32016
Class. Math.: 32S22, 53C35, 55N25
Keywords: hyperplane arrangement, local system, Milnor fiber
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.
[1] K. Aomoto & M. Kita, Hypergeometric Functions, (in Japanese), Springer-Verlag, 1994
[2] A. Beauville, Monodromie des systèmes différentiels linéaires à pôles simples sur la sphère de Riemann (d'après A. Bolibruch), Astérisque, 1993, p. 103-119
Numdam | Zbl 0796.34007
[3] A. Beilinson, J. Bernstein & P. Deligne, Faisceaux Pervers, Astérisque, Soc. Math. France, 1982, p. 5-171 Zbl 0536.14011
[4] A. Bolibrukh, “The Riemann-Hilbert problem”, Russian Math. Surveys 45 (1990), p. 1-58 MR 1069347 | Zbl 0706.34005
[5] D. Cohen & A. Suciu, “On Milnor fibrations of arrangements”, J. London Math. Soc. 51 (1995), p. 105-119 MR 1310725 | Zbl 0814.32007
[6] J. Damon, On the number of bounding cycles for nonlinear arrangements, Adv. Stud. Pure Math, 2000, p. 51-72 Zbl 0991.32016
[7] P. Deligne, Équations Différentielles à Points Singuliers Réguliers, Lect. Notes in Math. 163, Springer-Verlag, 1970 MR 417174 | Zbl 0244.14004
[8] A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag 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, Hyperplane arrangements, $M$-tame polynomials and twisted cohomology, NATO Science Series, Kluwer, 2003, p. 113-126 Zbl 1046.32003
[11] A. Dimca & A. Némethi, “Hypersurface complements, Alexander modules and monodromy”, Proceedings of the 7th Workshop on Real and Complex Singularities (Sao Carlos, 2002), to appear, preprint, math.AG/0201291, 2002
arXiv | MR 2087802 | Zbl 1067.14004
[12] A. Dimca & S. Papadima, “Equivariant chain complexes, twisted homology and relative minimality of arrangements”, e-print, math.AG/0305266, 2003
arXiv | MR 2060483
[13] H. Esnault, V. Schechtman & V. Viehweg, “Cohomology of local systems on the complement of hyperplanes”, Invent. Math. 109 (1992), p. 557-561
Article | MR 1176205 | Zbl 0788.32005
[14] H. Esnault & E. Viehweg, “Logarithmic de Rham complexes and vanishing theorems”, Invent. Math. 86 (1986), p. 161-194
Article | MR 853449 | Zbl 0603.32006
[15] I. M. Gelfand, “General theory of hypergeometric functions”, Soviet Math. Dokl. 33 (1986) MR 841131 | Zbl 0037.15302
[16] M. Kashiwara & P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss. 292, Springer-Verlag, 1994 MR 1299726 | Zbl 0709.18001
[17] T. Kohno, “Homology of a local system on the complement of hyperplanes”, Proc. Japan Acad., Ser. A 62 (1986), p. 144-147 MR 846350 | Zbl 0611.55005
[18] V. Kostov, Regular linear systems on $CP\sp 1$ and their monodromy groups, Astérisque, 1994, p. 259-283 Zbl 0814.34006
[19] A. Libgober, The topology of complements to hypersurfaces and nonvanishing of a twisted de Rham cohomology, AMS/IP Stud. Adv. Math., Amer. Math. Soc., 1997, p. 116-130 Zbl 0934.14009
[20] A. Libgober, Eigenvalues for the monodromy of the Milnor fibers of arrangements, Trends Math., Birkhäuser, 2002, p. 141-150 Zbl 1036.32019
[21] D. Massey, “Perversity, duality and arrangements in $\C^3$”, Topology Appl. 73 (1996), p. 169-179 MR 1416758 | Zbl 0867.32018
[22] P. Orlik & H. Terao, Arrangements of Hyperplanes, Grundlehren Math. Wiss. vol. 300, Springer-Verlag MR 1217488 | Zbl 0757.55001
[23] P. Orlik & H. Terao, Arrangements and Hypergeometric Integrals, MSJ Mem. 9, Math. Soc. Japan, 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 100 (1995), p. 93-102 MR 1344845 | Zbl 0849.32025
[25] A. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Adv. Ser. Math. Phys. 21, World Scientific, 1995 MR 1384760 | Zbl 0951.33001
[26] S. Yuzvinsky, “Cohomology of the Brieskorn-Orlik-Solomon algebras”, Comm. Algebra 23 (1995), p. 5339-5354 MR 1363606 | Zbl 0851.32027
[<L>13] H. Esnault, V. Schechtman & E. Viehweg, “Erratum: "Cohomology of local systems on the complement of hyperplanes"”, Invent. Math 112 (1993) MR 1213111 | Zbl 0794.32008
Daniel C. Cohen; Alexandru Dimca; Peter Orlik
Nonresonance conditions for arrangements
(Conditions de non-résonance pour les arrangements)
Annales de l'institut Fourier, 53 no. 6 (2003), p. 1883-1896
Article PDF | Reviews MR 2038782 | Zbl 1054.32016
Class. Math.: 32S22, 53C35, 55N25
Keywords: hyperplane arrangement, local system, Milnor fiber
Résumé - Abstract
Bibliography
[2] A. Beauville, Monodromie des systèmes différentiels linéaires à pôles simples sur la sphère de Riemann (d'après A. Bolibruch), Astérisque, 1993, p. 103-119
Numdam | Zbl 0796.34007
[3] A. Beilinson, J. Bernstein & P. Deligne, Faisceaux Pervers, Astérisque, Soc. Math. France, 1982, p. 5-171 Zbl 0536.14011
[4] A. Bolibrukh, “The Riemann-Hilbert problem”, Russian Math. Surveys 45 (1990), p. 1-58 MR 1069347 | Zbl 0706.34005
[5] D. Cohen & A. Suciu, “On Milnor fibrations of arrangements”, J. London Math. Soc. 51 (1995), p. 105-119 MR 1310725 | Zbl 0814.32007
[6] J. Damon, On the number of bounding cycles for nonlinear arrangements, Adv. Stud. Pure Math, 2000, p. 51-72 Zbl 0991.32016
[7] P. Deligne, Équations Différentielles à Points Singuliers Réguliers, Lect. Notes in Math. 163, Springer-Verlag, 1970 MR 417174 | Zbl 0244.14004
[8] A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag 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, Hyperplane arrangements, $M$-tame polynomials and twisted cohomology, NATO Science Series, Kluwer, 2003, p. 113-126 Zbl 1046.32003
[11] A. Dimca & A. Némethi, “Hypersurface complements, Alexander modules and monodromy”, Proceedings of the 7th Workshop on Real and Complex Singularities (Sao Carlos, 2002), to appear, preprint, math.AG/0201291, 2002
arXiv | MR 2087802 | Zbl 1067.14004
[12] A. Dimca & S. Papadima, “Equivariant chain complexes, twisted homology and relative minimality of arrangements”, e-print, math.AG/0305266, 2003
arXiv | MR 2060483
[13] H. Esnault, V. Schechtman & V. Viehweg, “Cohomology of local systems on the complement of hyperplanes”, Invent. Math. 109 (1992), p. 557-561
Article | MR 1176205 | Zbl 0788.32005
[14] H. Esnault & E. Viehweg, “Logarithmic de Rham complexes and vanishing theorems”, Invent. Math. 86 (1986), p. 161-194
Article | MR 853449 | Zbl 0603.32006
[15] I. M. Gelfand, “General theory of hypergeometric functions”, Soviet Math. Dokl. 33 (1986) MR 841131 | Zbl 0037.15302
[16] M. Kashiwara & P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss. 292, Springer-Verlag, 1994 MR 1299726 | Zbl 0709.18001
[17] T. Kohno, “Homology of a local system on the complement of hyperplanes”, Proc. Japan Acad., Ser. A 62 (1986), p. 144-147 MR 846350 | Zbl 0611.55005
[18] V. Kostov, Regular linear systems on $CP\sp 1$ and their monodromy groups, Astérisque, 1994, p. 259-283 Zbl 0814.34006
[19] A. Libgober, The topology of complements to hypersurfaces and nonvanishing of a twisted de Rham cohomology, AMS/IP Stud. Adv. Math., Amer. Math. Soc., 1997, p. 116-130 Zbl 0934.14009
[20] A. Libgober, Eigenvalues for the monodromy of the Milnor fibers of arrangements, Trends Math., Birkhäuser, 2002, p. 141-150 Zbl 1036.32019
[21] D. Massey, “Perversity, duality and arrangements in $\C^3$”, Topology Appl. 73 (1996), p. 169-179 MR 1416758 | Zbl 0867.32018
[22] P. Orlik & H. Terao, Arrangements of Hyperplanes, Grundlehren Math. Wiss. vol. 300, Springer-Verlag MR 1217488 | Zbl 0757.55001
[23] P. Orlik & H. Terao, Arrangements and Hypergeometric Integrals, MSJ Mem. 9, Math. Soc. Japan, 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 100 (1995), p. 93-102 MR 1344845 | Zbl 0849.32025
[25] A. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Adv. Ser. Math. Phys. 21, World Scientific, 1995 MR 1384760 | Zbl 0951.33001
[26] S. Yuzvinsky, “Cohomology of the Brieskorn-Orlik-Solomon algebras”, Comm. Algebra 23 (1995), p. 5339-5354 MR 1363606 | Zbl 0851.32027
[<L>13] H. Esnault, V. Schechtman & E. Viehweg, “Erratum: "Cohomology of local systems on the complement of hyperplanes"”, Invent. Math 112 (1993) MR 1213111 | Zbl 0794.32008
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310