A note on stable complex structures on real vector bundles over manifolds

Let M be an n-dimensional closed oriented smooth manifold with n≡0mod8, ξ be a real vector bundle over M. Suppose that ξ admits a stable complex structure over the (n−1)-skeleton of M. Then the necessary and sufficient conditions for ξ to admit a stable complex structure over M are given in terms of the characteristic classes of ξ and M. As an application, we obtain the criteria to determine which real vector bundle over 8-dimensional manifold admits a stable complex structure.