Newton polyhedra and the Bergman kernel
type: Item
instance of: scholarly article
title: Newton polyhedra and the Bergman kernel
zbMATH Open document ID: 1047.32004
author: Joe Kamimoto
published in: Mathematische Zeitschrift
publication date: 2004-07-01T00:00:00Z
review text: The author studies the structure of singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Assuming that the boundary \(\partial\Omega\) of \(\Omega\in\mathbb C^{n+1}\) is \(C^\infty\)-smooth, the Bergman kernel \(B(z)\) of \(\Omega\) takes the form near a boundary point \(p\): \[ B(z) =\frac{\Phi(w,p)}{\rho^{2+2/d_F}(\log(1/\rho))^{m_F-1}}, \] where \((w,p)\) is some polar coordinates on a nontangential cone \(\Lambda\) with apex at \(p\) and \(\rho\) means the distance from the boundary. Here \(\Phi\) admits some asymptotic expansion with respect to the variables \(\rho^{1/m}\) and \(\log(1/\rho)\) as \(\rho\to 0\) on \(\Lambda\). The values of \(d_F > 0\), \(m_F\in\mathbb Z_+\) and \(m\in\mathbb N\) are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of \(\Phi\) as \(\rho\to 0\) on \(\Lambda\) is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel. The analysis is based on some integral formula for the Bergman kernel due to Haslinger. The Fourier transform gives a clear connection between the Bergman space of \(\Omega\) and an associated weighted Bergman space \(H_\tau\) in \(\mathbb C^n\) with a parameter \(\tau\). As a result, the Bergman kernel of \(\Omega\) is expressed as a superposition of the reproducing kernel of \(H_\tau\) by using the Laplace transform with respect to \(\tau\). From the Haslinger formula, the problem is reduced to the analysis of a Laplace integral of the following form: \(I(\tau)=\int_{\mathbb R_+^n}e^{-\tau f(x)}\varphi(x)\,dx,\) \(\tau>0\), where the phase \(f\) is associated with the defining function of \(\Omega\). In order to investigate the boundary behavior of the Bergman kernel, the author analyzes the behavior of the integral \(I(\tau)\) at infinity. The essential idea of the analy…
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 32T25
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 2078059
zbMATH Keywords: singularities of the Bergman kernel
zbMATH Keywords: pseudoconvex domains of finite type
zbMATH Keywords: Newton polyhedra
zbMATH Keywords: Szegö kernel
zbMATH Keywords: Laplace integral
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: Fourier analysis
zbMATH Keywords: toric varieties
zbMATH Keywords: toric resolutions
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
full work available at URL: s00209-003-0554-7
OpenAlex ID: W2063647849
DOI: 10.1007/S00209-003-0554-7
description: scientific article; zbMATH DE number 2078059