The Bergman Kernel on Monomial Polyhedra

摘要

In order to understand the Bergman kernel for a complex domain Ω in Cn at z close to the boundary δΩ, we usually insert the biholomorphic image of a polydisc D centered at z in Ω to generate the upper bound for the Bergman kemel on Ω: KΩ(z, z) ≤ Kd(z, z) = 1/Vol(D) On the other hand, Catlin [3] showed by using a δ estimate that, on a finite type pseudoconvex domain Ω in C2, there exists a polydisc D such that KΩ(z, z) ≥ c. 1/Vol(D); the same formula was later shown by McNeal [8] on convex domains in Cn. A question anses. Are polydiscs enough to describe the Bergman kernel for smooth bounded domains.