The Bergman and Szeg\H o kernels in a bounded domain $\Omega\subset \mathbb C^n$ are the reproducing kernels for the holomorphic functions in $L^2(\Omega,dV)$ and $L^2(\partial \Omega,d\sigma)$, respectively, where $dV$ denotes the standard Lebesgue measure in $\bC^n$ and $d\sigma$ a surface measure on the boundary $\partial\Omega$. Their restrictions to the diagonal are known to have asymptotic expansions of the form:
$$K_B\sim \frac{\phi_B}{\rho^{n+1}}+\psi_B\log\rho,\quad K_S\sim \frac{\phi_S}{\rho^{n}}+\psi_S\log\rho,$$
where $\phi_B,\phi_S,\psi_B,\psi_S\in C^\infty(\overline{\Omega})$ and $\rho>0$ is a defining equation for $\Omega$. The functions $\phi_B,\phi_S,\psi_B,\psi_S$ encode a wealth of information about the biholomorphic geometry of $\Omega$ and its boundary $\partial \Omega$. In this talk, we will discuss this in the context of bounded strictly pseudoconvex domains in $\mathbb C^2$ and pay special attention to the lowest order invariants in the log term and a strong form of a conjecture of Ramadanov.
