Note on the filtrations of the K-theory

Abstract

Let X be a (colimit of) smooth algebraic variety over a subfield k of C. Let K_alg⁰(X) (resp. K_top⁰(X(C))) be the algebraic (resp. topological) K-theory of k (resp. complex) vector bundles over X (resp. X(C))). When K_alg⁰(X) $\cong$ K_top⁰(X(C)), we study the differences of its three (gamma, geometrical and topological) filtrations. In particular, we consider in the cases X = BG for algebraic group G over algebraically closed fields k, and X = G_k/T_k the twisted form of flag varieties G/T for non-algebraically closed field k.