Holomorphic K-Theory, Algebraic Co-cycles, and Loop Groups

Abstract

In this paper we study the “holomorphic K -theory” of a projective variety. This K theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a related theory was considered in [11]. This theory is built out of studying algebraic bundles over a variety up to “algebraic equivalence”. In this paper we will give calculations of this theory for “flag like varieties” which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors in [6], we will show that there is a rational isomorphism of graded rings between holomorphic K theory and the appropriate “morphic cohomology” groups, defined in [7] in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the “symmetrized loop group” ΩU(n)/Σn where the symmetric group Σn 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 use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K theory by inverting the Bott class, then rationally this is isomorphic to topological K theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K theory, and show that these obstructions vanish for generalized flag manifolds.