  We define a renormalized characteristic class for Einstein asymptotically
complex hyperbolic (ACHE) manifolds of dimension 4: for any such manifold, the
polynomial in the curvature associated to the characteristic class
euler-3signature is shown to converge. This extends a work of Burns and Epstein
in the Kahler-Einstein case. This extends a work of Burns and Epstein in the
Kahler-Einstein case. We also define a new global invariant for any
3-dimensional pseudoconvex CR manifold, by a renormalization procedure of the
eta invariant of a sequence of metrics which approximate the CR structure.
Finally, we get a formula relating the renormalized characteristic class to the
topological number euler-3signature and the invariant of the CR structure
arising at infinity.
