Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
type: Item
instance of: scholarly article
title: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
zbMATH Open document ID: 1120.32021
author: Bo Berndtsson
publication date: 2007-04-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0505469
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10187
review text: The main result of this paper states that if \(\phi\) is a plurisubharmonic function in a pseudoconvex domain~\(D\) in the product space \(\mathbb{C}^{k}_{t} \times \mathbb{C}^{n}_{z}\), and \(K_{t}(z,w)\) is the Bergman kernel function for the slice domain \(\{ z: (t,z)\in D\}\) with respect to the weight function \(e^{-\phi(t,z)}\), then the function \(\log K_{t}(z,z)\) is plurisubharmonic in~\(D\) (or else identically equal to \(-\infty\)). \textit{F.~Maitani} and \textit{H.~Yamaguchi} [Math. Ann. 330, No. 3, 477--489 (2004; Zbl 1077.32006)] previously obtained the simplest case when \(\phi=0\) and \(n=1\). The author gives two proofs and some interesting applications.
reviewed by: Harold P. Boas
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 5145724
zbMATH Keywords: Bergman spaces
zbMATH Keywords: Lelong number
zbMATH Keywords: plurisubharmonic function
MaRDI profile type: Publication
OpenAlex ID: W2108523452
arXiv ID: math/0505469
cites work: Entropy jumps in the presence of a spectral gap
cites work: Prekopa's theorem and Kiselman's minimum principle for plurisubharmonic functions
cites work: On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
cites work: Holomorphic approximation and estimates for the \({\bar \partial}\)-equation on strictly pseudoconvex nonsmooth domains
cites work: On the strict concavity of the harmonic radius in dimension \(N\geq 3\)
cites work: Santaló's inequality on \(\mathbb{C}^n\) by complex interpolation
cites work: On Berndtsson's generalization of Prékopa's theorem
cites work: Estimations $\mathrm{L}^2$ pour l'opérateur $\bar \partial$ d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète
cites work: \(L^ 2\) estimates and existence theorems for the \(\partial\)-operator
cites work: The partial Legendre transformation for plurisubharmonic functions
cites work: Densité des fonctions plurisousharmoniques
cites work: Attenuating the singularities of plurisubharmonic functions
cites work: Variation of Bergman metrics on Riemann surfaces
cites work: Q5682159
cites work: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents
cites work: Sous-ensembles analytiques d'ordre fini ou infini dans $C^n$
cites work: Variations of pseudoconvex domains over \({\mathbb{C}}^ n\)
DOI: 10.5802/AIF.2223
published in: Annales de l’institut Fourier
description: scientific article