Analytic singularities of the Bergman kernel for tubes
type: Item
instance of: scholarly article
title: Analytic singularities of the Bergman kernel for tubes
zbMATH Open document ID: 1016.32014
DOI: 10.1215/S0012-7094-01-10835-1
author: Nicholas Hanges
author: Gábor Francsics
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: Let \(\Omega\subset\mathbb R^n\) be a bounded, convex, and open set with real analytic boundary. Let \(T_\Omega\subset\mathbb C^n\) be the tube with base \(\Omega\), and let \(\mathcal B(z,w)\) be the Bergman kernel of \(T_\Omega\). If \(\Omega\) is strongly convex, then \(\mathcal B(z,w)\) is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. The authors show that off-diagonal singularities are precisely described by the characteristic lines. These lines are contained in the boundary of \(T_\Omega\), and they are projections of the Treves curves. These curves are symplectic invariants that are determined by the CR structure of the boundary of \(T_\Omega\). Note that Treves curves exist only when \(\Omega\) has at least one weakly convex boundary point. A brief discussion of the formula of L.~Boutet de Monvel for \(\mathcal B(z,w)\) is given. The precise version of the stationary phase formula that the authors need is proved. Using the method of Pólya, it is proved that the zeros of the entire function \(\mathcal N(\eta)=\int_{-\infty}^{+\infty}e^{2(\eta s-s^m)} ds\), \(m\geq 2\) is even, which are closest to the origin are simple ones.
Mathematics Subject Classification ID: 32T27
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 35H10
zbMATH DE Number: 1820829
zbMATH Keywords: Bergman kernel for tubes
zbMATH Keywords: analytic singularities away from the boundary diagonal
zbMATH Keywords: characteristic lines
zbMATH Keywords: symplectic geometry
zbMATH Keywords: Treves curves
zbMATH Keywords: stationary phase formula
zbMATH Keywords: Gevrey regularity
zbMATH Keywords: algorithm of Pólya
MaRDI profile type: Publication
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full work available at URL: s0012-7094-01-10835-1
OpenAlex ID: W2085200063
description: scientific article; zbMATH DE number 1820829