Exotic characteristic classes of quaternionic bundles

For aC ^∞ quaternionic vector bundle, the odd-dimensional real Chern classes vanish, and this allows for a construction of secondary (exotic) characteristic classes associated with a pair of quaternionic structures of a given complex vector bundle. This construction is then applied to obtain exotic characteristic classes associated with an automorphismβ of the holomorphic tangent bundle of a Kähler manifold. These results are the complex analoga of those given for the higher order Maslov classes in V2].