  Using symmetrized Grassmannians we give an algebraic geometric presentation,
in the level of classifying spaces, of the Chern character and its relation to
Chern classes. This allows one to define, for any projective variety $X$, a
Chern character map from the "holomorphic K-theory of X to its morphic
cohomology (introduced by Friedlander and Lawson). The holomorphic K-theory of
$X$, introduced by Lawson, Lima-Filho and Michelsohn and also by Friedlander
and Walker, is defined in terms a group-completion of the space of algebraic
morphisms from X into BU. It has been further studied by the authors in a
companion paper. Holomorphic K-theory sits between algebraic K-theory and
topological K-theory in the same way that morphic cohomology sits between
motivic cohomology and ordinary cohomology. Our constructions provide a bridge
between these two worlds. We also realize Chern classes in the case where X is
smooth, and establish a universal relation between the Chern character and the
Chern classes. We use classical constructions with algebraic cycles and
infinite symmetric products of projective spaces. The latter can be seen as the
classifying space for motivic cohomology, and under this perspective our
constructions are essentially motivic.
