Semi-topological K-theory using function complexes

Abstract

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 K semi (X) which is equipped with maps K alg (X)→ K semi (X) , K semi (X)→ K top (X an ) whose composition is the natural map from the algebraic K-theory of X to the topological K-theory of the underlying analytic space X an of X. The theory K semi (X) defined and studied here is equivalent (when X is projective and weakly normal) to the so-called “holomorphic K-theory”, K hol (X) , of projective varieties, which is studied by Cohen and Lima-Filho. We give an explicit description of K 0 semi( X) in terms of K 0( X), a description of K q semi(−) in terms of K 0 semi(−) for projective varieties, a Poincaré duality theorem for projective varieties, and a computation of K semi (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 X an.