Analytic singularities of the Bergman kernel for tubes

Abstract

Let Ω⊂ℝⁿ be a bounded, convex, and open set with real analytic boundary. Let T_Ω⊂ℂⁿ be the tube with base Ω, and let $\mathcal{B}$ be the Bergman kernel of T_Ω. 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 T_Ω, 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 T_Ω. Note that Treves curves exist only when Ω has at least one weakly convex boundary point.