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We present a renormalized Gauss-Bonnet formula for a strictly pseudoconvex manifold with a complete K\"{a}hler metric given by a globally defined potential function near the boundary. When the metric is asymptotically Einstein, the boundary contribution in the formula is explicitly written down in terms of the pseudo-hermitian geometry of the boundary and is shown to be a global CR invariant. The CR invariant generalizes the Burns-Epstein invariant on 3-dimensional CR manifolds.

Renormalized Chern-Gauss-Bonnet formula for complete Kähler-Einstein metrics