Analytic singularities of the Bergman kernel for tubes

Abstract

Let Ω⊂ℝn be a bounded, convex, and open set with real analytic boundary. Let TΩ⊂ℂn be the tube with base Ω, and let $\mathcal{B}$ be the Bergman kernel of TΩ. If Ω is strongly convex, then $\mathcal{B}$ 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 characteristic lines. These lines are contained in the boundary of TΩ, and they are projections of the Treves curves. These curves are symplectic invariants that are determined by the CR (Cauchy-Riemann) structure of the boundary of TΩ. Note that Treves curves exist only when Ω has at least one weakly convex boundary point.