Semi-topological K-theory using function complexes

The semi-topological K-theory K∗semi(X) of a quasi-projective complex algebraic variety X is based on the notion of algebraic vector bundles modulo algebraic equivalence. This theory is given as the homotopy groups of an infinite loop space Ksemi(X) which is equipped with maps Kalg(X)→Ksemi(X), Ksemi(X)→Ktop(Xan) whose composition is the natural map from the algebraic K-theory of X to the topological K-theory of the underlying analytic space Xan of X. The theory Ksemi(X) defined and studied here is equivalent (when X is projective and weakly normal) to the so-called “holomorphic K-theory”, Khol(X), of projective varieties, which is studied by Cohen and Lima-Filho. We give an explicit description of K0semi(X) in terms of K0(X), a description of Kqsemi(−) in terms of K0semi(−) for projective varieties, a Poincaré duality theorem for projective varieties, and a computation of Ksemi(X) whenever X is a product of projective spaces or a smooth complete curve. For X a smooth quasi-projective variety, there are natural Chern class maps from K∗semi(X) to morphic cohomology compatible with similarly defined Chern class maps from algebraic K-theory to motivic cohomology and compatible with the classical Chern class maps from topological K-theory to the singular cohomology of Xan.