Extension of a positive quasi-plurisuperharmonic current
type: Item
instance of: scholarly article
title: Extension of a positive quasi-plurisuperharmonic current
zbMATH Open document ID: 1241.32028
author: Noureddine Ghiloufi
author: Khalifa Dabbek
published in: Annales Mathématiques Blaise Pascal
publication date: 2009-11-11T00:00:00Z
full work available at URL: https://eudml.org/doc/10581
review text: This paper is concerned with the problem of extension of a positive current through a ``small'' obstacle. Namely, let \(A\) be a closed subset of an open set \(\Omega\subset\mathbb{C}^n\) and let \(T\) be a positive current of bidimenstion \((p,p)\) on \(\Omega\setminus A\). The problem is to give geometric conditions on the obstacle \(A\) and the current \(T\) which insure that the current \(T\) can be extended to positive current on \(\Omega\) with additional properties. This problem has its origin in the following remarkable result of \textit{E. Bishop} [Mich. Math. J. 11, 289--304 (1964; Zbl 0143.30302)]: if \(A\) is an proper analytic subset of \(\Omega\) and \(X\) is an analytic subset of \(\Omega\setminus A\) of pure dimension and locally finite volume near the points of \(A\), then the closure of \(X\) is an analytic subset of \(\Omega\). In [Invent. Math. 66, 361--376 (1982; Zbl 0488.58002)], \textit{H. Skoda} proved the following deep generalization of Bishop's result: if \(A\) is a (proper) analytic subset of \(\Omega\) and \(T\) is a closed positive current on \(\Omega\setminus A\) with locally finite mass near the points of \(A\), then its trivial extension \(\widetilde T:=\mathbf{1}_{\Omega\setminus A}T\) is a closed positive current. Later, \textit{H. El Mir} [Acta Math. 153, 1--45 (1984; Zbl 0557.32003)] gave an interesting generalization of Skoda's result to the case when the obstacle \(A\) is a locally complete pluripolar subset of \(\Omega\), i.e. locally \(A\) is given as the set where a plurisubharmonic function takes the value \(-\infty\). When the current is not assumed to be closed, \textit{T.-C. Dinh} and \textit{N. Sibony} [Manuscr. Math. 123, No. 3, 357--371 (2007; Zbl 1128.32020)] gave the following generalization of El Mir's theorem: if \(T\) is positive and quasi-plurisurharmonic (e.g. \(dd^cT\leq 0\)), then \(dd^cT\) has locally finite mass near the points of \(A\) and \(d\widetilde d^cT-dd^c\widetilde T\) is a positive current support…
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32U40
Mathematics Subject Classification ID: 32V20
zbMATH DE Number: 5629442
zbMATH Keywords: plurisubharmonic function
zbMATH Keywords: pluripolar sets
zbMATH Keywords: positive current
zbMATH Keywords: extension
zbMATH Keywords: slicing
reviewed by: Ahmed Zeriahi
MaRDI profile type: Publication
OpenAlex ID: W2233368016
cites work: Q3918395
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DOI: 10.5802/AMBP.268
description: scientific article; zbMATH DE number 5629442