Comparison of the Beilinson-Chern Classes With the Chern-Cheeger-Simons Classes

  • Jean-Luc Brylinski
Part of the Progress in Mathematics book series (PM, volume 172)


The Chern-Cheeger-Simons (CCS) classes of a vector bundle with connection belong to the group ofdifferential characters of[Chee-S], and depend on the choice of a connection. For a projective complex manifold, we introduce a smaller group, the group ofrestricted differential characters, which contains the CCS classes of holomorphic vector bundles equipped with a connection compatible with the complex structure.We construct a map from this group to a Deligne cohomology group, and show that unintroduce logarithmic resticted differential character as a receptacle of the CCS classes for algebraic vector bundles equipped with a connection with logarithmic singularities, and related the CCS classes to the Beilinson-Chern classes in the logarithmic context. From this we deduce that for a flat vector bundleEover a quasi-projective algebraic manifoldX, the Beilinson-Chern classes given are the images of the Chern-Cheeger-Simons classes under some canonical map(thereby extending results of Bloch [BI] and Soulé[S]).


Exact Sequence Vector Bundle Line Bundle Chern Class Holomorphic Vector Bundle 
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  1. [Be]
    A. A. BeilinsonHigher regulators and values of L-functionsJ.Soviet Math.30(1985), 2036–2070MATHCrossRefGoogle Scholar
  2. [BI]
    S. BlochApplications of the dilogarithm in algebraic K-theory and algebraic geometryInt. Symp. on Alg. Geometry, Kyoto (1977), 103–114Google Scholar
  3. [Chee-S]
    J. Cheeger and J. SimonsDifferential characters and geometric invariantsLecture Notes in Math.1167(1980), Springer Verlag, 50–80MathSciNetGoogle Scholar
  4. [Cher-S]
    S. S. Chern and J. SimonsCharacteristic forms and geometric invariantsAnn. Math.99(1974), 48–69MathSciNetMATHCrossRefGoogle Scholar
  5. [D1]
    P. DeligneEquations différentielles à points singuliers réguliersLecture Notes in Math.163(1970),15–27MathSciNetGoogle Scholar
  6. [D2]
    P. DeligneThéorie de Hodge IIPubl. Math. IHES40(1971), 5–58MathSciNetMATHGoogle Scholar
  7. [D3]
    P. DeligneLe symbole modéréPubl. Math. IHES73(1991), 147–181MathSciNetMATHGoogle Scholar
  8. [D-H-Z]
    J. Dupont, R. Hain and S. ZuckerRegulators and characteristic classes of flat bundlespreprint Aarhus Univ. (1992)Google Scholar
  9. [E]
    H. EsnaultCharacteristic classes of flat bundlesTopology27(1988), 323–352MathSciNetMATHCrossRefGoogle Scholar
  10. [E-V]
    H. Esnault and E. ViehwegDeligne-Beilinson cohomologyin Beilinson’s Conjectures and Values of L-Functions, Perspectives in Math. Acad. Press (1988), 43–92Google Scholar
  11. [G]
    P. Griffiths, The extension problem in complex analysisIII. Em-beddings with positive normal bundleAmer. J. Math.88(1966), 366–446MATHCrossRefGoogle Scholar
  12. [H]
    H. HironakaResolution of singularities over a field of character-istic zeroAnn. Math.79(1964), 109–326MathSciNetMATHCrossRefGoogle Scholar
  13. [K]
    B. KostantQuantization and Unitary Representationsin Lec-ture Notes in Math.170(1970), 87–208MathSciNetGoogle Scholar
  14. [M]
    J. MorganThe algebraic topology of smooth algebraic varietiesPubl. Math. IHES48(1978), 137–204MATHGoogle Scholar
  15. [S]
    C. SouléConnexions et classes caractéristiques de BeilinsoninAlgebraic K-Theory and Algebraic Number Theory, Contemp. Math.83(1989), Amer. Math. Soc., 349–376CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Jean-Luc Brylinski
    • 1
  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA

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