Nombres de Lelong généralisés, théorèmes d'intégralité et d'analyticité. (Generalized Lelong numbers, integrability and analyticity theorems)
type: Item
instance of: scholarly article
title: Nombres de Lelong généralisés, théorèmes d'intégralité et d'analyticité. (Generalized Lelong numbers, integrability and analyticity theorems)
zbMATH Open document ID: 0629.32011
DOI: 10.1007/BF02392558
author: Jean-Pierre Demailly
published in: Acta Mathematica
publication date: 1987-01-01T00:00:00Z
review text: Let X be a complex Stein space, T a closed positive current of bidimension (p,p) on X and \(\phi:X\to [-\infty,+\infty [\) an exhaustive plurisubharmonic function. The author's generalized Lelong number \(\nu\) (T,\(\phi)\) is defined as the mass of the measure \(T\wedge (dd^ c\phi)^ p\) carried by the polar set \(\phi^{-1}(-\infty)\) and is obtained by means of the Monge-Ampère operator of \textit{E. Bedford} and \textit{B. A. Taylor} [ibid. 149, 1-40 (1982; Zbl 0547.32012)]. \(\nu\) (T,\(\phi)\) generalizes the classical \textit{P. Lelong} [``Plurisubharmonic functions and positive differential forms'' (1969; Zbl 0195.116)] and C. O. Kiselman's numbers. The author establishes that \(\nu\) (T,\(\phi)\) depends only on the behaviour of \(\phi\) in a neighbourhood of the poles. The use of \(\nu\) (T,\(\phi)\) allows him to obtain very simple proofs of classical results on Lelong numbers, e.g. that these numbers are invariant with respect to local coordinate transformations [cf. \textit{Y. T. Siu}, Invent. Math. 27, 53-156 (1974; Zbl 0289.32003)] and also on \textit{P. Thie}'s [Math. Ann. 172, 269-312 (1967; Zbl 0158.328)] theorem showing that the Lelong number of an analytic set X coincides to the algebraic multiplicity of Y at x. Finally, the author obtains a generalization of Siu's theorem on the analyticity of the level sets associated to Lelong numbers, his result containing as a particular case a recent theorem of C. O. Kiselman on directional Lelong numbers.
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32C30
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 31C15
zbMATH DE Number: 4023690
zbMATH Keywords: integrability
zbMATH Keywords: potential
zbMATH Keywords: complex Stein space
zbMATH Keywords: current
zbMATH Keywords: plurisubharmonic function
zbMATH Keywords: generalized Lelong number
zbMATH Keywords: analyticity
reviewed by: Petru Caraman
MaRDI profile type: Publication
cites work: A new capacity for plurisubharmonic functions
cites work: Algebraic values of meromorphic maps
cites work: Q5601113
cites work: Sur les nombres de Lelong associés à l'image directe d'un courant positif fermé
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Q3215513
cites work: Densité des fonctions plurisousharmoniques
cites work: Intégration sur un ensemble analytique complexe
cites work: Fonctions entières (\(n\) variables) et fonctions plurisousharmoniques d'ordre fini dans \(C^ n\)
cites work: Projektionen analytischer Mengen
cites work: Holomorphe und meromorphe Abbildungen komplexer Räume
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: The Lelong number of a point of a complex analytic set
full work available at URL: https://doi.org/10.1007/bf02392558
OpenAlex ID: W2320379890
description: scientific article; zbMATH DE number 4023690