The Szeg\"o kernel for certain non-pseudoconvex domains in C²

We consider the Szeg\"o kernel for domains \Omega in C^2 given by \Omega = {(z,w): Im w > b(Re z)} where b is a non-convex quartic polynomial with positive leading coefficient. Such domains are not pseudoconvex. We describe the subset of \bar{\Omega} \times \bar{\Omega} on which the kernel and all its derivatives are finite. In particular, we show that there are points off the diagonal of the boundary at which the Szeg\"o kernel is infitie as well as points on the diagonal at which it is finite.