RALPH L. COHEN - Holomorphic K-theory, Lawson's Chow - cohomology, and loop groups

In this talk I will describe a program to study holomorphic bundles over smooth projective varieties using K-theoretic techniques. The ``holomorphic K-theory" I will describe will be defined in terms of spaces of holomorphic mappings from a variety to Grassmannians or to loop groups. I will describe the structure of this theory, how its characteristic classes take values in Lawson's Chow cohomology groups, and how the analogue of Bott periodicity holds for this theory, and is proved using the geometry of loop groups. I will describe results that relate holomorphic K-theory to both algebraic K-theory and topological K-theory, and describe some sample calculations. In doing so I will give a geometric description of the Chern character in terms of the ``symmetrized loop group'', , where the symmetric group acts on U(n) by conjugation. Finally, I will show how this is used to prove a theorem (joint with Lima-Filho) stating that modulo torsion, all Lawson cohomology classes (and in particular all cohomology classes represented by algebraic cocycles) are realized as (algebraic) Chern classes of holomorphic bundles.