Volume renormalization for complete Einstein–Kähler metrics

For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern–Gauss–Bonnet formula. Relations between renormalized volume and CR Q-curvature are also investigated.