  We show that many spin 6-manifolds have the homotopy type but not the
homeomorphism type of a Kaehler manifold. Moreover, for given Betti numbers,
there are only finitely many deformation types and hence topological types of
smooth complex projectve spin threefolds of general type. Finally, on a fixed
spin 6-manifold, the Chern numbers take on only finitely many values on all
possible Kaehler structures.
