Fefferman's expansion of the Bergman kernel

Remarks

The value of $\varphi$ on $\partial\Omega$ is an explicit expression in the first and second derivatives of $\rho$, built from the Levi form. Hörmander had found this leading behaviour before Fefferman. Fefferman's proof compares $\Omega$ near each boundary point with a model domain osculating it, and controls the error with estimates for the $\bar\partial$-Neumann problem. Boutet de Monvel and Sjöstrand later gave a microlocal proof that describes the kernel off the diagonal as well.

Example: for the unit ball $B \subset \C^n$, $K(z) = \dfrac{n!}{\pi^n (1 - \abs{z}^2)^{n+1}}$. With $\rho(z) = \abs{z}^2 - 1$ this is the expansion with $\varphi = n!/\pi^n$ and $\psi = 0$. The logarithmic term is in general nonzero, and its coefficient $\psi$ on $\partial\Omega$ carries geometric information about the boundary.

Strong pseudoconvexity is needed for the exponent $n + 1$. For the unit polydisc in $\C^2$, $K(z) = \pi^{-2}(1 - \abs{z_1}^2)^{-2}(1 - \abs{z_2}^2)^{-2}$. Let $\delta$ be the distance to the boundary. Near a point with $\abs{z_1} = 1$ and $\abs{z_2} < 1$, the kernel grows only like $\delta^{-2}$, not like $\delta^{-3}$.

Going deeper

For a student

The expansion is the analytic input of Fefferman's theorem on the extension of biholomorphisms, through the boundary behaviour of the Bergman metric. It shows that the Bergman projection and the Bergman metric of a strongly pseudoconvex domain look, near the boundary, like those of the ball. The coefficients $\varphi$ and $\psi$ restricted to $\partial\Omega$ are local invariants of the boundary under biholomorphic maps. This led Fefferman to a programme of computing them from the CR geometry of $\partial\Omega$.