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

Abstract

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 Soule[S]).