Hirzebruch–Milnor Classes and Steenbrink Spectra of Certain Projective Hypersurfaces

  • Laurentiu Maxim
  • Morihiko SaitoEmail author
  • Jörg Schürmann
Part of the Progress in Mathematics book series (PM, volume 319)


We show that the Hirzebruch–Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g., in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch–Milnor classes of complete intersections.


Irreducible Component Complete Intersection Hyperplane Arrangement Smooth Projective Variety Mixed Hodge Structure 
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The first named author is partially supported by NSF-1304999. The second named author is partially supported by Kakenhi 24540039. The third named author is supported by the SFB 878 “groups, geometry and actions.”


  1. [Al]
    P. Aluffi, Differential forms with logarithmic poles and Chern-Schwartz-MacPherson classes of singular varieties. C. R. Acad. Sci. Paris Sér. I Math. 329, 619–624 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [BaFuMa]
    P. Baum, W. Fulton, R. MacPherson, Riemann-Roch for singular varieties. Inst. Hautes Etudes Sci. Publ. Math. 45, 101–145 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Bri]
    E. Brieskorn, Sur les groupes de tresses, in Séminaire Bourbaki (1971/1972), Exp. 401. Lecture Notes in Mathematics, vol. 317 (Springer, Berlin, 1973), pp. 21–44Google Scholar
  4. [BrScYo]
    J.-P. Brasselet, J. Schürmann, S. Yokura, Hirzebruch classes and motivic Chern classes of singular spaces. J. Topol. Anal. 2, 1–55 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BuSa1]
    N. Budur, M. Saito, Multiplier ideals, V -filtration, and spectrum. J. Algebra Geom. 14, 269–282 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BuSa2]
    N. Budur, M. Saito, Jumping coefficients and spectrum of a hyperplane arrangement. Math. Ann. 347, 545–579 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [CaMaScSh]
    S.E. Cappell, L. Maxim, J. Schürmann, J.L. Shaneson, Characteristic classes of complex hypersurfaces. Adv. Math. 225, 2616–2647 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [dB]
    Ph. du Bois, Complexe de de Rham filtré d’une variété singulière. Bull. Soc. Math. Fr. 109, 41–81 (1981)Google Scholar
  9. [DCPr]
    C. De Concini, C. Procesi, Wonderful models of subspace arrangements. Sel. Math. N. Ser. 1, 459–494 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [De1]
    P. Deligne, Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970)Google Scholar
  11. [De2]
    P. Deligne, Théorie de Hodge, II. Inst. Hautes Etudes Sci. Publ. Math. 40, 5–57 (1971)CrossRefzbMATHGoogle Scholar
  12. [De3]
    P. Deligne, Le formalisme des cycles évanescents, in SGA7 XIII and XIV. Lecture Notes in Mathematics, vol. 340 (Springer, Berlin, 1973), pp. 82–115, 116–164.Google Scholar
  13. [De4]
    P. Deligne, Théorie de Hodge, III. Inst. Hautes Etudes Sci. Publ. Math. 44, 5–77 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [DiMaSaTo]
    A. Dimca, Ph. Maisonobe, M. Saito, T. Torrelli, Multiplier ideals, V -filtrations and transversal sections. Math. Ann. 336, 901–924 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [EsScVi]
    H. Esnault, V. Schechtman, E. Viehweg, Cohomology of local systems on the complement of hyperplanes. Invent. Math. 109, 557–561 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Fu]
    F. Fulton, Intersection Theory (Springer, Berlin, 1984)CrossRefzbMATHGoogle Scholar
  17. [GoPa]
    M. Goresky, W. Pardon, Chern classes of automorphic vector bundles. Inv. Math. 147, 561–612 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [Hi]
    F. Hirzebruch, Topological Methods in Algebraic Geometry (Springer, Berlin, 1966)CrossRefzbMATHGoogle Scholar
  19. [HiBeJu]
    F. Hirzebruch, T. Berger, R. Jung, Manifolds and Modular Forms. Aspects of Mathematics, vol. E20 (Friedrich Vieweg & Sohn, Braunschweig, 1992)Google Scholar
  20. [LiMa]
    A. Libgober, L. Maxim, Hodge polynomials of singular hypersurfaces. Michigan Math. J. 60, 661–673 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Mac]
    R.D. MacPherson, Chern classes for singular algebraic varieties. Ann. Math. (2) 100, 423–432 (1974)Google Scholar
  22. [MaSaSc1]
    L. Maxim, M. Saito, J. Schürmann, Hirzebruch-Milnor classes of complete intersections. Adv. Math. 241, 220–245 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [MaSaSc2]
    L. Maxim, M. Saito, J. Schürmann, Spectral Hirzebruch-Milnor classes of singular hypersurfaces. Preprint arXiv:1606.02218Google Scholar
  24. [Mi]
    J. Milnor, Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, vol. 61 (Princeton University Press, Princeton, NJ, 1968)Google Scholar
  25. [PaPr]
    A. Parusiński, P. Pragacz, Characteristic classes of hypersurfaces and characteristic cycles. J. Alg. Geom. 10, 63–79 (2001)MathSciNetzbMATHGoogle Scholar
  26. [Sa1]
    M. Saito, Modules de Hodge polarisables. Publ. RIMS Kyoto Univ. 24, 849–995 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Sa2]
    M. Saito, Mixed hodge modules. Publ. RIMS Kyoto Univ. 26, 221–333 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Sa3]
    M. Saito, Thom-Sebastiani theorem for Hodge modules. Preprint 2010Google Scholar
  29. [ScTeVa]
    V. Schechtman, H. Terao, A. Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors. J. Pure Appl. Algebra 100, 93–102 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [Sch]
    J. Schürmann, Characteristic classes of mixed hodge modules, in Topology of Stratified Spaces. MSRI Publications, vol. 58 (Cambridge University Press, Cambridge, 2011), pp. 419–471.Google Scholar
  31. [St1]
    J.H.M. Steenbrink, Mixed hodge structure on the vanishing cohomology, in Real and Complex Singularities. Proceedings of Ninth Nordic Summer School, Oslo, 1976 (Sijthoff and Noordhoff, Alphen aan den Rijn, 1977), pp. 525–563Google Scholar
  32. [St2]
    J.H.M. Steenbrink, The spectrum of hypersurface singularities. Astérisque 179–180, 163–184 (1989)MathSciNetzbMATHGoogle Scholar
  33. [Yo1]
    S. Yokura, A generalized Grothendieck-Riemann-Roch theorem for Hirzebruch’s χ y-characteristic and T y-characteristic. Publ. RIMS Kyoto Univ. 30, 603–610 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [Yo2]
    S. Yokura, On characteristic classes of complete intersections, in Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998). Contemporary Mathematics, vol. 241 (American Mathematical Society, Providence, RI, 1999), pp. 349–369Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Laurentiu Maxim
    • 1
  • Morihiko Saito
    • 2
    Email author
  • Jörg Schürmann
    • 3
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.RIMS Kyoto UniversityKyotoJapan
  3. 3.Mathematisches InstitutUniversität MünsterMünsterGermany

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