Chern classes of spherical CR manifolds

We first construct a closed spherical CR manifold of dimension at least five with non-trivial first Chern class in real coefficients. In particular, it admits no pseudo-Einstein contact forms. We next give a constraint on Chern classes of (not necessarily closed) spherical CR manifolds in real coefficients. Finally we obtain a topological obstruction to the existence of a spherical CR structure on a given co-oriented contact manifold.