A “relative” $K$-theory group for holomorphic or algebraic vector bundles on a compact or quasiprojective complex manifold is constructed, and Chern-Simons type characteristic classes are defined on this group in the spirit of Nadel. In the projective case, their coincidence with the Abel-Jacobi image of the Chern classes of the bundles is proved. Some applications to families of holomorphic bundles are given and two Riemann-Roch type theorems are proved for these classes.