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.
Olivier Biquard. Marc Herzlich. "A Burns-Epstein invariant for ACHE 4-manifolds." Duke Math. J. 126 (1) 53 - 100, 15 January 2005. https://doi.org/10.1215/S0012-7094-04-12612-0