Title: Holomorphic $K$-theory, algebraic co-cycles, and loop groups
Authors: Ralph L. Cohen and Paulo Lima-Filho
AMS Class: Primary: 55P91 ; Secondary 14C05, 19L47, 55N91
Addresses: Stanford University and Texas A&M University
Email addresses: ralph@math.stanford.edu and plfilho@math.tamu.edu
Text of abstract
In this paper we study the ``holomorphic $K$-theory" of a projective
variety. This theory 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 show that there is a rational
isomorphism between holomorphic $K$-theory and the appropriate
"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" $\om U(n)/\Sigma_n$ where the symmetric group $\Sigma_n$ 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.