  It is known that the Picard group of a complex manifold can be expressed as a
Deligne cohomology group. One may wonder if the same holds for the Picard group
of a smooth algebraic variety and Deligne-Beilinson cohomology but this is not
true, as already remarked by M. Saito. We explain how one has to modify the
latter, show that the Picard group can be expressed by absolute Hodge
cohomology, too, and introduce an intermediate object between Picard group and
usual Deligne-Beilinson cohomology group. Similarly as in the case of Deligne
cohomology one can relate line bundles with a regular connection to (modified)
Deligne-Beilinson cohomology. In order to take irregular connections into
account one has to change the definition of Deligne-Beilinson cohomology even
more.
