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Gauss-Manin connections for arrangements, III Formal connections


Authors: Daniel C. Cohen and Peter Orlik
Journal: Trans. Amer. Math. Soc. 357 (2005), 3031-3050
MSC (2000): Primary 32S22, 14D05, 52C35, 55N25
DOI: https://doi.org/10.1090/S0002-9947-04-03621-9
Published electronically: July 16, 2004
MathSciNet review: 2135734
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Abstract: We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal Gauss-Manin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to Gauss-Manin connection matrices.


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  • 1. K. Aomoto, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan 39 (1987), 191-208. MR 88f:32031
  • 2. K. Aomoto, M. Kita, Hypergeometric Functions (in Japanese), Springer-Verlag, Tokyo, 1994.
  • 3. D. Cohen, Cohomology and intersection cohomology of complex hyperplane arrangements, Adv. Math. 97 (1993), 231-266. MR 94a:32055
  • 4. D. Cohen, P. Orlik, Arrangements and local systems, Math. Res. Lett. 7 (2000), 299-316. MR 2001i:57040
  • 5. -, Some cyclic covers of complements of arrangements, Topology Appl. 118 (2002), 3-15. MR 2003h:32039
  • 6. -, Gauss-Manin connections for arrangements, I. Eigenvalues, Compositio Math. 136 (2003), 299-316. MR 2004a:32042
  • 7. -, Gauss-Manin connections for arrangements, II. Nonresonant weights, Amer. J. Math., to appear; math.AG/0207114.
  • 8. D. Cohen, A. Suciu, On Milnor fibrations of arrangements, J. London Math. Soc. 51 (1995), 105-119. MR 96e:32034
  • 9. P. Deligne, Equations Différentielles à Points Singuliers Réguliers, Lect. Notes in Math., vol. 163, Springer-Verlag, Berlin-New York, 1970. MR 54:5232
  • 10. H. Esnault, V. Schechtman, V. Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992), 557-561; Erratum, ibid. 112 (1993), 447. MR 93g:32051; MR 94b:32061
  • 11. M. Falk, H. Terao, $\beta$nbc-bases for cohomology of local systems on hyperplane complements, Trans. Amer. Math. Soc. 349 (1997), 189-202. MR 97g:52029
  • 12. I. Gelfand, General theory of hypergeometric functions, Soviet Math. Dokl. 33 (1986), 573-577. MR 87h:22012
  • 13. M. Goresky, R. MacPherson, Stratified Morse Theory, Ergeb. Math. Grenzgeb., vol. 14, Springer-Verlag, Berlin-New York, 1988. MR 90d:57039
  • 14. H. Kanarek, Gauss-Manin connection arising from arrangements of hyperplanes, Illinois J. Math. 44 (2000), 741-766. MR 2002m:14006
  • 15. J. Kaneko, The Gauss-Manin connection of the integral of the deformed difference product, Duke Math. J. 92 (1998), 355-379. MR 99h:32024
  • 16. S. Kobayashi, Differential geometry of complex vector bundles, Princeton Univ. Press, Princeton, NJ, 1987. MR 89e:53100
  • 17. P. Orlik, H. Terao, Arrangements of Hyperplanes, Grundlehren Math. Wiss., vol. 300, Springer-Verlag, Berlin, 1992. MR 94e:52014
  • 18. -, Arrangements and Hypergeometric Integrals, MSJ Mem., vol. 9, Math. Soc. Japan, Tokyo, 2001. MR 2003a:32048
  • 19. R. Randell, Lattice-isotopic arrangements are topologically isomorphic, Proc. Amer. Math. Soc. 107 (1989), 555-559. MR 90a:57032
  • 20. V. Schechtman, H. Terao, A. Varchenko, Cohomology of local systems and the Kac-Kazhdan condition for singular vectors, J. Pure Appl. Algebra 100 (1995), 93-102. MR 96j:32047
  • 21. V. Schechtman and A. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139-194. MR 93b:17067
  • 22. H. Terao, Moduli space of combinatorially equivalent arrangements of hyperplanes and logarithmic Gauss-Manin connections, Topology Appl. 118 (2002), 255-274. MR 2003e:32049
  • 23. A. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Adv. Ser. Math. Phys., vol. 21, World Scientific, River Edge, NJ, 1995. MR 99i:32029

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Additional Information

Daniel C. Cohen
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: cohen@math.lsu.edu

Peter Orlik
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: orlik@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03621-9
Keywords: Hyperplane arrangement, local system, Gauss-Manin connection
Received by editor(s): July 15, 2003
Published electronically: July 16, 2004
Additional Notes: The first author was partially supported by Louisiana Board of Regents grant LEQSF(1999-2002)-RD-A-01 and by National Security Agency grant MDA904-00-1-0038, and the second author was partially supported by National Security Agency grant MDA904-02-1-0019
Article copyright: © Copyright 2004 American Mathematical Society

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