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
χ−3τ is shown to converge. This extends a work of Burns
and Epstein in the Kähler-Einstein case
¶ We also define a new global invariant for any compact
3-dimensional strictly pseudoconvex Cauchy-Riemann (CR) manifold
by a renormalization procedure of the η-invariant of a
sequence of metrics that approximate the CR structure.
¶ Finally, we get a formula relating the renormalized characteristic
class to the topological number χ−3τ and the invariant of
the CR structure arising at infinity.