The Bergman Kernel and a Theorem of Tian

Abstract

Given a domain Ω in ℂn, the Bergman kernel is the kernel of the projection operator from L 2 (Ω) to the Hardy space A 2 (Ω). When the boundary of Ω is strictly pseudoconvex and smooth, Fefferman [2] gave a complete description of the asymptotic behavior of K(z, z) as z approaches the boundary. This work was then extended by Boutet de Monvel and Sjostrand [1] who showed that, for the same domains, a similar asymptotic expansion for K(z, w) holds off the diagonal. Moreover, they showed that the Bergman kernel is a Fourier integral operator with a complex phase function.