Nonresonance conditions for arrangements

Daniel C. Cohen[1]; Alexandru Dimca[2]; Peter Orlik[3]

  • [1] Louisiana State University, Department of Mathematics, Baton Rouge LA 70803 (USA)
  • [2] Université Bordeaux I, Laboratoire de Mathématiques Pures, 351 cours de la Libération, 33405 Talence Cedex (France)
  • [3] University of Wisconsin, Department of Mathematics, Madison WI 53706 (USA)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 6, page 1883-1896
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Cohen, Daniel C., Dimca, Alexandru, and Orlik, Peter. "Nonresonance conditions for arrangements." Annales de l’institut Fourier 53.6 (2003): 1883-1896. <http://eudml.org/doc/116087>.

@article{Cohen2003,
abstract = {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.},
affiliation = {Louisiana State University, Department of Mathematics, Baton Rouge LA 70803 (USA); Université Bordeaux I, Laboratoire de Mathématiques Pures, 351 cours de la Libération, 33405 Talence Cedex (France); University of Wisconsin, Department of Mathematics, Madison WI 53706 (USA)},
author = {Cohen, Daniel C., Dimca, Alexandru, Orlik, Peter},
journal = {Annales de l’institut Fourier},
keywords = {hyperplane arrangement; local system; Milnor fiber; vanishing theorem},
language = {eng},
number = {6},
pages = {1883-1896},
publisher = {Association des Annales de l'Institut Fourier},
title = {Nonresonance conditions for arrangements},
url = {http://eudml.org/doc/116087},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Cohen, Daniel C.
AU - Dimca, Alexandru
AU - Orlik, Peter
TI - Nonresonance conditions for arrangements
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 6
SP - 1883
EP - 1896
AB - 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.
LA - eng
KW - hyperplane arrangement; local system; Milnor fiber; vanishing theorem
UR - http://eudml.org/doc/116087
ER -

References

top
  1. K. Aomoto, M. Kita, Hypergeometric Functions, (in Japanese), (1994), Springer-Verlag, Tokyo Zbl1229.33001
  2. A. Beauville, Monodromie des systèmes différentiels linéaires à pôles simples sur la sphère de Riemann (d'après A. Bolibruch), Séminaire Bourbaki, Vol. 1992/93 216, Exp. No. 765, 4 (1993), 103-119 Zbl0796.34007
  3. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux Pervers, Analysis and topology on singular spaces, I (Luminy, 1981) 100 (1982), 5-171, Soc. Math. France, Paris Zbl0536.14011
  4. A. Bolibrukh, The Riemann-Hilbert problem, Russian Math. Surveys 45 (1990), 1-58 Zbl0706.34005MR1069347
  5. D. Cohen, A. Suciu, On Milnor fibrations of arrangements, J. London Math. Soc. 51 (1995), 105-119 Zbl0814.32007MR1310725
  6. J. Damon, On the number of bounding cycles for nonlinear arrangements, Arrangements--Tokyo 1998 27 (2000), 51-72, Kinokuniya, Tokyo Zbl0991.32016
  7. P. Deligne, Équations Différentielles à Points Singuliers Réguliers, 163 (1970), Springer-Verlag, Berlin-New York Zbl0244.14004MR417174
  8. A. Dimca, Singularities and Topology of Hypersurfaces, Springer-Verlag, New York Zbl0753.57001MR1194180
  9. A. Dimca, Sheaves in Topology Zbl1043.14003MR2050072
  10. A. Dimca, Hyperplane arrangements, M -tame polynomials and twisted cohomology, Commutative Algebra, Singularities and Computer Algebra Vol. 115 (2003), 113-126, Kluwer Zbl1046.32003
  11. A. Dimca, A. Némethi, Hypersurface complements, Alexander modules and monodromy, (2002) Zbl1067.14004MR2087802
  12. A. Dimca, S. Papadima, Equivariant chain complexes, twisted homology and relative minimality of arrangements, (2003) Zbl1059.32007MR2060483
  13. H. Esnault, V. Schechtman, V. Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992), 557-561 Zbl0788.32005MR1176205
  14. H. Esnault, E. Viehweg, Logarithmic de Rham complexes and vanishing theorems, Invent. Math. 86 (1986), 161-194 Zbl0603.32006MR853449
  15. I. M. Gelfand, General theory of hypergeometric functions, Soviet Math. Dokl. 33 (1986) Zbl0037.15302MR841131
  16. M. Kashiwara, P. Schapira, Sheaves on Manifolds, 292 (1994), Springer-Verlag, Berlin Zbl0709.18001MR1299726
  17. T. Kohno, Homology of a local system on the complement of hyperplanes, Proc. Japan Acad., Ser. A 62 (1986), 144-147 Zbl0611.55005MR846350
  18. V. Kostov, Regular linear systems on C P 1 and their monodromy groups, Complex analytic methods in dynamical systems (Rio de Janeiro, 1992) No 222 (1994), 259-283 Zbl0814.34006
  19. A. Libgober, The topology of complements to hypersurfaces and nonvanishing of a twisted de Rham cohomology, Singularities and complex geometry (Beijing, 1994) 5 (1997), 116-130, Amer. Math. Soc., Providence, RI Zbl0934.14009
  20. A. Libgober, Eigenvalues for the monodromy of the Milnor fibers of arrangements, Trends in Singularities (2002), 141-150, Birkhäuser Zbl1036.32019
  21. D. Massey, Perversity, duality and arrangements in 3 , Topology Appl. 73 (1996), 169-179 Zbl0867.32018MR1416758
  22. P. Orlik, H. Terao, Arrangements of Hyperplanes, vol. 300, Springer-Verlag, Berlin Zbl0757.55001MR1217488
  23. P. Orlik, H. Terao, Arrangements and Hypergeometric Integrals, 9 (2001), Math. Soc. Japan, Tokyo Zbl0980.32010MR1814008
  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), 93-102 Zbl0849.32025MR1344845
  25. A. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, 21 (1995), World Scientific, River Edge Zbl0951.33001MR1384760
  26. S. Yuzvinsky, Cohomology of the Brieskorn-Orlik-Solomon algebras, Comm. Algebra 23 (1995), 5339-5354 Zbl0851.32027MR1363606
  27. H. Esnault, V. Schechtman, E. Viehweg, Erratum: "Cohomology of local systems on the complement of hyperplanes", Invent. Math 112 (1993) Zbl0794.32008MR1213111

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.