Holomorphic K-theory, algebraic co-cycles, and loop groups

In this paper we study the "holomorphic K-theory" of a projective variety, which is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory was introduced by Lawson, Lima-Filho and Michelsohn, and also by Friedlander and Walker, and a related theory was considered by Karoubi. Using the Chern character studied by the authors in a companion paper, we give a rational isomorphism between holomorphic $K$-theory and the "morphic cohomology" defined by Lawson and Friedlander. In doing so, we describe a geometric model for rational morphic cohomology groups in terms of algebraic maps from the variety to the "symmetrized loop group" Loops(U(n)/S_n) where the symmetric group S_n acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K-theory by inverting the Bott class, then it is rationally isomorphic to topological K-theory. Finally we produce explicit obstructions to periodicity in holomorphic K-theory, and show that these obstructions vanish for generalized flag manifolds.