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Singularities of admissible normal functions

(with an appendix by Najmuddin Fakhruddin)

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Abstract

In a recent paper, M. Green and P. Griffiths used R. Thomas’ work on nodal hypersurfaces to sketch a proof of the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using Morihiko Saito’s mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equivalent to the Hodge conjecture. Several of the intermediate results, including a relative version of the weak Lefschetz theorem for perverse sheaves, are of independent interest.

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References

  1. Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves. Vol. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267. Springer, New York (1985)

    Google Scholar 

  2. Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I, Luminy, 1981. Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982)

    Google Scholar 

  3. Bloch, S.: Algebraic cycles and values of L-functions. J. Reine Angew. Math. 350, 94–108 (1984)

    MATH  MathSciNet  Google Scholar 

  4. de Cataldo, A.M.A., Migliorini, L.: A remark on singularities of primitive cohomology classes (2007). arXiv:0711.1307

  5. Deligne, P.: Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  6. Deligne, P.: Décompositions dans la catégorie dérivée. In: Motives, Seattle, WA, 1991. Proc. Sympos. Pure Math., vol. 55, pp. 115–128. Am. Math. Soc., Providence (1994)

    Google Scholar 

  7. Dimca, A., Saito, M.: Vanishing cycles of one parameter smoothings (2008)

  8. Green, M., Griffiths, P.: Algebraic cycles and singularities of normal functions. In: Algebraic Cycles and Motives. Vol. 1. London Math. Soc. Lecture Note Ser., vol. 343, pp. 206–263. Cambridge Univ. Press, Cambridge (2007)

    Google Scholar 

  9. Groupes de monodromie en géométrie algébrique. II. Lecture Notes in Mathematics, vol. 340. Springer, Berlin (1973). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Dirigé par P. Deligne et N. Katz

  10. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

    MATH  Google Scholar 

  11. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  12. Hironaka, H.: Triangulations of algebraic sets. In: Algebraic Geometry, Proc. Sympos. Pure Math., vol. 29, Humboldt State Univ., Arcata, Calif., 1974, pp. 165–185. Am. Math. Soc., Providence (1975)

    Google Scholar 

  13. Katz, N.M.: Perversity and exponential sums. II. Estimates for and inequalities among A-numbers. In: Barsotti Symposium in Algebraic Geometry, Abano Terme, 1991. Perspect. Math., vol. 15, pp. 205–252. Academic Press, San Diego (1994)

    Google Scholar 

  14. Katz, N.M.: Rigid Local Systems, Annals of Mathematics Studies, vol. 139. Princeton University Press, Princeton (1996)

    MATH  Google Scholar 

  15. Kaup, L.: Zur Homologie projektiv algebraischer Varietäten. Ann. Sc. Norm. Super. Pisa (3) 26, 479–513 (1972)

    MATH  MathSciNet  Google Scholar 

  16. Lojasiewicz, S.: Triangulation of semi-analytic sets. Ann. Sc. Norm. Super. Pisa (3) 18, 449–474 (1964)

    MATH  MathSciNet  Google Scholar 

  17. Mumford, D.: Abelian Varieties. Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Published for the Tata Institute of Fundamental Research, Bombay (1970)

  18. Ngô, B.C.: Le lemme fondamental pour les algebres de lie (2008). arXiv:0801.0446

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

    Google Scholar 

  20. Saito, M.: Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. 24(6), 849–995 (1989) 1988

    Article  Google Scholar 

  21. Saito, M.: Mixed Hodge modules. Publ. Res. Inst. Math. Sci. 26(2), 221–333 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Saito, M.: Hodge conjecture and mixed motives. I. In: Complex Geometry and Lie Theory, Sundance, UT, 1989. Proc. Sympos. Pure Math., vol. 53, pp. 283–303. Am. Math. Soc., Providence (1991)

    Google Scholar 

  23. Saito, M.: Some remarks on the Hodge type conjecture. In: Motives, Seattle, WA, 1991. Proc. Sympos. Pure Math., vol. 55, pp. 85–100. Am. Math. Soc., Providence (1994)

    Google Scholar 

  24. Saito, M.: Admissible normal functions. J. Algebraic Geom. 5(2), 235–276 (1996)

    MATH  MathSciNet  Google Scholar 

  25. Saito, M.: Generalized Thomas hyperplane sections and relations between vanishing cycles (2008). arXiv:0806.1461

  26. Thomas, R.P.: Nodes and the Hodge conjecture. J. Algebraic Geom. 14(1), 177–185 (2005)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada, V6T 1Z2

    Patrick Brosnan

  2. Department of Mathematics, College of Liberal Arts & Sciences, University of Iowa, 14 MLH, Iowa City, IA, 52242, USA

    Hao Fang

  3. Department of Mathematics, Penn State Altoona, 3000 Ivyside Park, Altoona, PA, 16601, USA

    Zhaohu Nie

  4. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Gregory Pearlstein

Authors
  1. Patrick Brosnan
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  2. Hao Fang
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  3. Zhaohu Nie
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  4. Gregory Pearlstein
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Corresponding author

Correspondence to Patrick Brosnan.

Additional information

P. Brosnan’s research was supported in part by an NSERC discovery grant.

H. Fang’s research was supported in part by NSF grant number DMS 0606721.

G. Pearlstein’s research was supported in part by NSF grant number DMS 0703956.

N. Fakhruddin

School of Mathematics, Tata Institue of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India

e-mail: naf@math.tifr.res.in

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Brosnan, P., Fang, H., Nie, Z. et al. Singularities of admissible normal functions. Invent. math. 177, 599–629 (2009). https://doi.org/10.1007/s00222-009-0191-9

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  • DOI: https://doi.org/10.1007/s00222-009-0191-9

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