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Hodge genera of algebraic varieties, II

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Abstract

We study the behavior of Hodge-genera under algebraic maps. We prove that the motivic \({\chi^c_y}\) -genus satisfies the “stratified multiplicative property”, which shows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic version of the Riemann–Hurwitz formula. We also study the monodromy contributions to the \({\chi_y}\) -genus of a family of compact complex manifolds, and prove an Atiyah–Meyer type formula in the algebraic and analytic contexts. This formula measures the deviation from multiplicativity of the \({\chi_y}\) -genus, and expresses the correction terms as higher-genera associated to the period map; these higher-genera are Hodge-theoretic extensions of Novikov higher-signatures to analytic and algebraic settings. Characteristic class formulae of Atiyah–Meyer type are also obtained by making use of Saito’s theory of mixed Hodge modules.

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References

  1. Arapura D.: The Leray spectral sequence is motivic. Invent. Math. 160, 567–589 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Atiyah, M.F.: The signature of fiber bundles. In: Global analysis (papers in Honor of K. Kodaira), pp. 73–84, Univ. Tokyo Press, Tokyo (1969)

  3. Atiyah M.F., Singer I.M.: The index of elliptic operators on compact manifolds. Bull. Am. Math. Soc. 69, 422–433 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  4. Banagl M., Cappell S.E., Shaneson J.L.: Computing twisted signatures and L-classes of stratified spaces. Math. Ann. 326, 589–623 (2003)

    MATH  MathSciNet  Google Scholar 

  5. Baum P., Fulton W., MacPherson M.: Riemann–Roch for singular varieties. Publ. Math. I.H.E.S. 45, 101–145 (1975)

    MATH  MathSciNet  Google Scholar 

  6. Borisov L.: Libgober: higher elliptic genera. Math. Res. Lett. 15(3), 511–520 (2008)

    MATH  MathSciNet  Google Scholar 

  7. Brasselet, P., Schürmann, J., Yokura, S.: Hirzebruch classes and motivic Chern classes of singular spaces, math.AG/0503492

  8. Cappell S.E., Maxim L.G., Shaneson J.L.: Euler characteristics of algebraic varieties. Comm. Pure Appl. Math. 61(3), 409–421 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cappell S.E., Maxim L.G., Shaneson J.L.: Hodge genera of algebraic varieties, I. Comm. Pure Appl. Math. 61(3), 422–449 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cappell S.E., Libgober A., Maxim L.G., Shaneson J.L.: Hodge genera and characteristic classes of complex algebraic varieties. Electron. Res. Announc. Math. Sci. 15, 1–7 (2008)

    MATH  MathSciNet  Google Scholar 

  11. Cappell, S.E., Maxim, L., Schürmann, J., Shaneson, J.L., Atiyah–Meyer formulae for intersection homology genera (in preparation)

  12. Cappell S.E., Shaneson J.L.: Stratifiable maps and topological invariants. J. Am. Math. Soc. 4(3), 521–551 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cappell S.E., Shaneson J.L.: Genera of algebraic varieties and counting of lattice points. Bull. Am. Math. Soc. (N.S.) 30(1), 62–69 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Carlson J., Muller-Stach S., Peters C.: Period Mappings and Period Domains. Cambridge Studies in Advanced Mathematics vol. 85. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  15. Chern S.S., Hirzebruch F., Serre J.-P.: On the index of a fibered manifold. Proc. Am. Math. Soc. 8, 587–596 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  16. Danilov V.I., Khovanskii A.G.: Newton polyhedra and an algorithm for computing Hodge–Deligne numbers. Bull. Am. Math. Soc. 30(1), 62–69 (1994)

    Article  Google Scholar 

  17. Deligne, P.: Théorie de Hodge, II, III. Publ. Math. IHES 40, 44 (1972, 1974)

  18. Deligne P.: Equation différentielles a point singular régulier, Lecture Notes in Mathematics, vol. 163. Springer, Berlin (1970)

    Google Scholar 

  19. Dimca A., Lehrer G.I.: Purity and equivariant weight polynomials, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., 9, pp. 161–181. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  20. Dimca A.: Sheaves in Topology, Universitext. Springer, Berlin (2004)

    Google Scholar 

  21. Fulton W.: Introduction to toric varieties, Annals of Mathematics Studies, vol. 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton (1993)

    Google Scholar 

  22. Fulton W.: Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2. Springer, Berlin (1998)

    Google Scholar 

  23. Goresky M., MacPherson R.: Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 14. Springer, Berlin (1988)

    Google Scholar 

  24. Phillip A., Phillip A.: Periods of integrals on algebraic manifolds, III. Publ. Math. I.H.E.S 38, 125–180 (1970)

    MATH  Google Scholar 

  25. Hirzebruch F.: Topological Methods in Algebraic Geometry. Springer, New York (1966)

    MATH  Google Scholar 

  26. Hirzebruch, F., Berger, T., Jung, R. (1992) Manifolds and modular forms, Aspects of Mathematics, E20. Friedr. Vieweg & Sohn, Braunschweig

  27. Kamber F., Tondeur P.H.: Flat manifolds, Lecture Notes in Mathematics, No. 67. Springer, Berlin (1968)

    Google Scholar 

  28. Kleiman, S.: The enumerative theory of singularities. In: Holm, P. (ed) Real and Complex Singularities, pp. 298–384. Sijthoff and Noordhoff (1976)

  29. Kodaira K.: A certain type of irregular algebraic surfaces. J. Anal. Math. 19, 207–215 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  30. Levy R.: Riemann–Roch theorem for complex spaces. Acta Math. 158, 149–188 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  31. MacPherson R.: Chern classes for singular algebraic varieties. Ann. Math. 100, 423–432 (1974)

    Article  MathSciNet  Google Scholar 

  32. Maxim, L.G., Schürmann, J.: Hodge-theoretic Atiyah–Meyer formulae and the stratified multiplicative property, Singularities I, pp. 145–166, Contemp. Math., 474, Amer. Math. Soc., Providence (2008)

  33. Meyer, W.: Die Signatur von lokalen Koeffizientensystemen und Faserbündeln, Bonner Mathematische Schriften 53, Universität Bonn (1972)

  34. Navarro Aznar, V.: Sur les structures de Hodge mixtes associées aux cycles évanescents, Hodge Theory (Sant Cugat, 1985), 143–153, Lecture Notes in Math., 1246, Springer, Berlin (1987)

  35. Navarro Aznar V.: Sur la théorie de Hodge–Deligne. Invent. Math. 90, 11–76 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ochanine, S.: Genres elliptiques équivariants, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986), 107–122, Lecture Notes in Math., 1326, Springer, Berlin (1988)

  37. Peters, C., Steenbrink, J.: Mixed Hodge structures, A Series of Modern Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics), 52. Springer, Berlin (2008)

  38. Saito M.: Modules de Hodge polarisables. Publ RIMS 24, 849–995 (1988)

    Article  MATH  Google Scholar 

  39. Saito M.: Mixed Hodge modules. Publ. RIMS 26, 221–333 (1990)

    Article  MATH  Google Scholar 

  40. Saito, M.: Introduction to mixed Hodge modules, Actes du Colloque de Théorie de Hodge (Luminy, 1987), Astérisque 179–180, 10, 145–162 (1989)

  41. Saito M.: Decomposition theorem for proper Kähler morphisms. Tohoku Math. J. 42, 127–148 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  42. Saito M.: Mixed Hodge complexes on algebraic varieties. Math. Ann. 316, 283–331 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  43. Schürmann J.: Topology of singular spaces and constructible sheaves, Monografie Matematyczne, 63. Birkhäuser, Basel (2003)

    Google Scholar 

  44. Schürmann J., Yokura S.: A survey of characteristic classes of singular spaces, in Singularity theory, pp. 865–952. World Scientific, Hackensack (2007)

    Google Scholar 

  45. Schürmann, J.: private communication

  46. Shaneson, S.: Characteristic classes, lattice points and Euler–MacLaurin formulae. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994), pp. 612–624. Birkhäuser, Basel (1995)

  47. Steenbrink J.: Intersection forms for quasihomogeneous singularities. Compos. Math. 34, 211–223 (1977)

    MATH  MathSciNet  Google Scholar 

  48. Totaro B.: Chern numbers for singular varieties and elliptic homology. Ann. Math. 151, 757–792 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  49. Yokura S.: On Cappell–Shaneson’s homology L-classes of singular algebraic varieties. Trans. Am. Math. Soc. 347, 1005–1012 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  50. Yokura, S.: A singular Riemann–Roch for Hirzebruch characteristics, Singularities Symposium– Lojasiewicz 70 (Kraków, 1996; Warsaw, 1996), pp. 257–268, Banach Center Publ., 44, Polish Acad. Sci., Warsaw (1998)

  51. Zucker S.: Hodge theory with degenerating coefficients: L 2-cohomology in the Poincaré-metric. Ann. Math. 109, 415–476 (1979)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Courant Institute, New York University, New York, NY, 10012, USA

    Sylvain E. Cappell & Laurentiu G. Maxim

  2. Department of Mathematics, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Anatoly Libgober

  3. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Julius L. Shaneson

Authors
  1. Sylvain E. Cappell
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  2. Anatoly Libgober
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  3. Laurentiu G. Maxim
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  4. Julius L. Shaneson
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Correspondence to Laurentiu G. Maxim.

Additional information

A. Libgober was partially supported by an NSF grant. S. Cappell and J. Shaneson were partially supported by grants from DARPA. L. Maxim was partially supported by a grant from the NYU Research Challenge Fund and a PSC-CUNY Research Award.

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Cappell, S.E., Libgober, A., Maxim, L.G. et al. Hodge genera of algebraic varieties, II. Math. Ann. 345, 925–972 (2009). https://doi.org/10.1007/s00208-009-0389-6

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  • DOI: https://doi.org/10.1007/s00208-009-0389-6

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