Analytic regularity for the Bergman kernel

Let Ω ⊂ R2 be a bounded, convex and open set with real analytic boundary. Let TΩ ⊂ C2 be the tube with base Ω, and let B be the Bergman kernel of TΩ. If Ω is strongly convex, then B is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation, we relate the off diagonal points where analyticity fails to the Treves curves. These curves are symplectic invariants which are determined by the CR structure of the boundary of TΩ. Note that Treves curves exist only when Ω has at least one weakly convex boundary point.