Global invariants for strongly pseudoconvex varieties with isolated singularities: Bergman functions
type: Item
instance of: scholarly article
title: Global invariants for strongly pseudoconvex varieties with isolated singularities: Bergman functions
zbMATH Open document ID: 1076.32024
published in: Mathematical Research Letters
publication date: 2005-04-05T00:00:00Z
review text: Let \(M\) be a strongly pseudoconvex \(n\)-dimensional manifold which is a resolution of a strongly pseudoconvex variety \(V\) with only isolated singularities. Let \(F\) be the set of all holomorphic \(n\)-forms \(\phi\) on \(M\) such that \(| \int_M\phi\wedge\overline{\phi}\;| \;<\infty\). Let \(\{\omega_j\}\) be a complete orthonormal basis of \(F\) and \(K(z, \overline{\omega}):=\sum \omega_j(z)\wedge\overline{\omega_j(w)}\). \(K(z, \overline{z})\) is a \(2n\)-form on \(M\), the so called Bergman kernel of \(M\). The Bergman kernel \(K_0 (z, \overline{z})\) vanishing on the exceptional set \(E\) of \(M\) is defined replacing \(F\) by \(F_0\), the set of all \(L^2\) integrable holomorphic \(n\)-forms vanishing on \(E\). The Bergman function \(B_M\) is defined to be \(K_0/K\). Pushing \(B_M\) down one obtains a Bergman function \(B_V\) on \(V\) which is a biholomorphic invariant of \(V\) and vanishes precisely on the singularities of \(V\). This Bergman function is able to distinguish the CR structures on the boundaries of \(V\). As an example \(V=V(xy-z^2)\) is treated.
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 32S45
Mathematics Subject Classification ID: 32T15
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 32V15
zbMATH DE Number: 2152378
zbMATH Keywords: Bergman function
zbMATH Keywords: isolated singularity
zbMATH Keywords: CR structure
author: Stephen Yau
MaRDI profile type: Publication
full work available at URL: mrl.2004.v11.n6.a8
OpenAlex ID: W2039617074
DOI: 10.4310/MRL.2004.V11.N6.A8
description: scientific article; zbMATH DE number 2152378