Comparing K-theories for complex varieties

The semi-topological K-theory of a complex variety was defined in a recent paper by the authors, with the expectation that it would prove to be a theo ry lying "part way" between the algebraic K-theory of the variety and the t opological K-theory of the associated analytic space, and thus would share properties with each of these other theories. In this paper, we realize the se expectations by proving among other results that (1) the algebraic K-the ory with finite coefficients and the semi-topological K-theory with finite coefficients coincide on all projective complex varieties, (2) semi-topolog ical K-theory and topological K-theory agree on certain types of generalize d flag varieties, and (3) (assuming a result asserted by Cohen and Lima-Fil ho) the semi-topological K-theory of any smooth projective variety becomes isomorphic to the topological K-theory of the underlying analytic space onc e the Bott element is inverted. To illustrate the utility of our results, w e observe that a new proof of the Quillen-Lichtenbaum conjecture for smooth , complete curves is obtained as a corollary.