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Comparison of the Beilinson-Chern classes with the Chern-Cheeger-Simons classes. (English) Zbl 0915.57013

Brylinski, Jean-Luc (ed.) et al., Advances in geometry. Boston, MA: Birkhäuser. Prog. Math. 172, 95-105 (1999).
The group of restricted differential characters of a projective complex manifold is introduced. It contains the Chern-Cheeger-Simons (CCS) classes of holomorphic vector bundles endowed with an almost complex connection. A map from this group to a Deligne cohomology group is also constructed and it is proved that, under this map, the CCS classes go to the Beilinson-Chern classes. The author introduces logarithmic restricted differential characters “as a receptacle of the CCS classes” for algebraic vector bundles endowed with a connection with logarithmic singularities and relate the CCS classes to the Beilinon-Chern classes in the logarithmic context. It follows that for a flat vector bundle over a quasi-projective algebraic manifold, the Beilinson classes are the images of the CCS classes under some canonical map, so that some results of S. Bloch [Proc. Int. Symp. Algebraic Geometry, Kyoto 1977, 103-114 (1977; Zbl 0416.18016)] and C. Soulé [Contemp. Math. 83, 349-376 (1989; Zbl 0695.14003)] are extended.
For the entire collection see [Zbl 0905.00030].

MSC:

57R20 Characteristic classes and numbers in differential topology
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
58A14 Hodge theory in global analysis
14F25 Classical real and complex (co)homology in algebraic geometry
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