Comparison of the Beilinson-Chern Classes With the Chern-Cheeger-Simons Classes
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]).
KeywordsExact Sequence Vector Bundle Line Bundle Chern Class Holomorphic Vector Bundle
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- [BI]S. BlochApplications of the dilogarithm in algebraic K-theory and algebraic geometryInt. Symp. on Alg. Geometry, Kyoto (1977), 103–114Google Scholar
- [D-H-Z]J. Dupont, R. Hain and S. ZuckerRegulators and characteristic classes of flat bundlespreprint Aarhus Univ. (1992)Google Scholar
- [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