The log term of the Szegö kernel
type: Item
instance of: scholarly article
title: The log term of the Szegö kernel
zbMATH Open document ID: 1072.32014
DOI: 10.1215/S0012-7094-04-12526-6
author: Zhiqin Lu
author: Gang Tian
published in: Duke Mathematical Journal
publication date: 2005-02-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1098892273
review text: The setting for this article is a positive Hermitian line bundle~\((L,h)\) over a compact complex manifold~\(M\) of dimension~\(n\). The authors study the asymptotic expansion of the Szegő kernel function for the unit circle bundle of the dual bundle~\(L^*\). They show that for a generic metric~\(h\), the logarithmic term of the Szegő kernel is not zero. When \(M\)~is complex projective space \(\mathbb{C}P^n\) and \(L\)~is the hyperplane bundle, they prove a local rigidity theorem: namely, if \(h\)~is the standard metric whose curvature is the Fubini-Study metric~\(\omega\), then for any metric~\(h'\) sufficiently close to~\(h\) for which the logarithmic term of the Szegő kernel of the corresponding unit circle bundle is zero, there is an automorphism~\(f\) of \(\mathbb{C}P^n\) such that the curvature of~\(h'\) pulled back by~\(f\) equals~\(\omega\). Moreover, the authors prove a corresponding global rigidity theorem when \(n=1\). They conjecture that the global rigidity holds for arbitrary~\(n\); this is an analogue for the Szegő kernel of the Ramadanov conjecture for the Bergman kernel.
reviewed by: Harold P. Boas
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 2139675
zbMATH Keywords: Ramadanov conjecture
zbMATH Keywords: pseudoconvex manifolds
zbMATH Keywords: projective space
MaRDI profile type: Publication
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Wikidata QID: Q125920272
description: scientific article; zbMATH DE number 2139675