Newton polyhedra and the Bergman kernel

Abstract

The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domain Ω in ℂn+1, the Bergman kernel B(z) of Ω takes the form near a boundary point p: \({{ B(z)= \frac{{\Phi(w,\rho)}}{{\rho^{{2+2/d_F}} (\hbox{{log}}(1/\rho))^{{m_F-1}}}}, }}\) where (w,ρ) is some polar coordinates on a nontangential cone Λ with apex at p and ρ means the distance from the boundary. Here Φ admits some asymptotic expansion with respect to the variables ρ1/m and log(1/ρ) as ρ→0 on Λ. The values of dF>0, mF∈ℤ+ and m∈ are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of Φ as ρ→0 on Λ is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szego kernel.