Local Topology of Reducible Divisors

  • Alexandru Dimca
  • Anatoly Libgober
Conference paper
Part of the Trends in Mathematics book series (TM)


We show that the universal abelian cover of the complement to a germ of a reducible divisor on a complex space Y with isolated singularity is (dimY − 2)-connected provided that the divisor has normal crossings outside of the singularity of Y. We apply this result to obtain a vanishing property for the cohomology of local systems of rank one and also study vanishing in the case of local systems of higher rank.


Fundamental Group Spectral Sequence Homotopy Type Normal Crossing Hyperplane Arrangement 
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  1. [1]
    H. Cartan, S. Eilenberg, Homological Algebra. Princeton University Press, Princeton, NJ, 1956.zbMATHGoogle Scholar
  2. [2]
    D. Cohen, A. Dimca and P. Orlik, Nonresonance conditions for arrangements, Ann. Institut Fourier 53 (2003), 1883–1896.MathSciNetGoogle Scholar
  3. [3]
    J. Damon, Critical points of affine multiforms on the complements of arrangements, Singularity Theory, ed. J. W. Bruce and D. Mond, London Math. Soc. Lect. Notes 263 (1999), CUP, 25–53.Google Scholar
  4. [4]
    A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer Verlag, 1992.Google Scholar
  5. [5]
    A. Dimca, Sheaves in Topology, Universitext, Springer Verlag, 2004.Google Scholar
  6. [6]
    A. Dimca, M. Saito, Some consequences of perversity of vanishing cycles, Ann.Inst. Fourier, Grenoble 54(2004), 1769–1792.Google Scholar
  7. [7]
    H. Esnault, E. Viehweg, Logarithmic de Rham complexes and vanishing theorems, Invent. Math. 86 (1986), 161–194.CrossRefMathSciNetGoogle Scholar
  8. [8]
    W. Fulton, Introduction to Toric Varieties, Annals of Math, Studies 131, Princeton University Press, 1993.Google Scholar
  9. [9]
    M. Goresky, R. MacPherson, Stratified Morse Theory. Springer Verlag.Google Scholar
  10. [10]
    M. Goresky, R. MacPherson, Intersection homology II, Invent. Math. 71 (1983), 77–129.CrossRefMathSciNetGoogle Scholar
  11. [11]
    A. Grothendieck, Cohomologie locale des faisceaux cohérents et Théorèmes de Lefschetz locaux et globaux, SGA2, North-Holland, 1968.Google Scholar
  12. [12]
    H. Hamm, Lokale topologische Eigenschaften komplexer Räume, Math. Ann. 191 (1971), 235–252.CrossRefMathSciNetGoogle Scholar
  13. [13]
    H. Hamm, Zum Homotopietyp Steinscher Räume, J. Reine Angew. Math. 338 (1983), 121–135.MathSciNetGoogle Scholar
  14. [14]
    H. Hamm, Lê Dung Trang, Local generalization of Lefschetz-Zariski theorem, J. Reine Angew. Math. 389 (1988), 157–189.MathSciNetGoogle Scholar
  15. [15]
    R. Hartshorne, Algebraic Geometry, GTM 52, Springer 1977.Google Scholar
  16. [16]
    M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss., vol. 292, Springer-Verlag, Berlin, 1994.Google Scholar
  17. [17]
    M. Kato, Y. Matsumoto, On the connectivity of the Milnor fiber of a holomorphic function at a critical point. Manifolds — Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), pp. 131–136. Univ. Tokyo Press, Tokyo, 1975.Google Scholar
  18. [18]
    Lê Dung Trang, K. Saito, The local π1 of the complement to a hypersurface with normal crossings in codimension 1 is abelian. Ark. Mat. 22 (1984), no. 1, 1–24.CrossRefMathSciNetGoogle Scholar
  19. [19]
    Lê, D.T., Complex analytic functions with isolated singularities, J. Algebraic Geometry 1 (1992), 83–100.Google Scholar
  20. [20]
    A. Libgober, On the homology of finite abelian coverings. Topology Appl. 43 (1992), no. 2, 157–166.CrossRefMathSciNetGoogle Scholar
  21. [21]
    A. Libgober, Hodge decomposition of Alexander invariants. Manuscripta Math. 107 (2002), no. 2, 251–269.CrossRefMathSciNetGoogle Scholar
  22. [22]
    A. Libgober, Isolated non normal crossings, in Real and Complex Singularities, eds. M. Ruas and T. Gaffney, Contemporary Mathematics, 354, 2004, 145–160.Google Scholar
  23. [23]
    J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematical Studies, 61, Princeton University Press, Princeton, 1968.Google Scholar
  24. [24]
    M. Nori, Zariski conjecture and related problems, Ann. Sci. Ecole Norm. Sup. (4), 16, (1983), 305–344.MathSciNetGoogle Scholar
  25. [25]
    P. Orlik, H. Terao, Arrangements of Hyperplanes, Grundlehren Math. Wiss., vol. 300, Springer-Verlag, Berlin, 1992.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Alexandru Dimca
    • 1
  • Anatoly Libgober
    • 2
  1. 1.Laboratoire J. A. Dieudonné, UMR CNRS 6621Université de Nice Sophia-AntipolisNice Cedex 2France
  2. 2.Department of MathematicsUniversity of Illinois at ChicagoChicagoUSA

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