### An algebraic geometric realization of the Chern character, by Ralph L. Cohen and Paulo Lima-Filho

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. With this we define, for any projective variety X, a Chern
character map ch : K^{-i}_{hol}(X) \to \prod_* L^*H^{2*-i}(X)\otimes Q 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. For this 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.

Ralph L. Cohen <ralph@math.stanford.edu>

Paulo Lima-Filho <plfilho@math.tamu.edu>