  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.
