The Monge-Ampère operator and slicing of closed positive currents
type: Item
instance of: scholarly article
title: The Monge-Ampère operator and slicing of closed positive currents
zbMATH Open document ID: 1005.32023
DOI: 10.1007/BF02921809
author: H. El Mir
author: Hèdi Ben Messaoud
published in: The Journal of Geometric Analysis
publication date: 2000-12-11T00:00:00Z
review text: Let \(k\leq p\leq n\) and \(R\) a current of bidimension \((p,p)\) in the polydisk \(\Delta^n= \Delta^k\times \Delta^{n-k}\) in \(\mathbb{C}^n\). Let \(\alpha\) be a measurable, bounded function with compact support in \(\mathbb{C}^k\), such that \(\int_{\mathbb{C}^k}\alpha= 1\). The slice of \(R\) at \(a\in\Delta^k\), denoted \(\langle R,\pi,a\rangle_\alpha\), where \(\pi(z)= (z_1,\dots, z_k)= z'\), is the limit as \(\varepsilon\to 0\) of \[ R\wedge \pi^*\Biggl({1\over \varepsilon^{2k}} \alpha\Biggl({z'- a\over\varepsilon}\Biggr)\cdot {1\over 4^k k!} (dd^c|z'|^2)^k\Biggr), \] when this exists. This agrees with the definition of Federer when \(\alpha\) is the characteristic function of the unit ball in \(\mathbb{C}^k\), and with Harvey and Shiffman for a smooth compactly supported function. The geometric idea of the slice is that when \(R\) is the current of integration on an analytic set \(X\) in \(\Delta^n\), then outside a countable union of analytic sets in \(\Delta^k\) the slice is given by the integration on the fiber \(X\cap \pi^{-1}(a)\) (proved in section 4 of the paper). Let \(T\) be a positive, closed current of bidimension \((p,p)\) in a neighborhood of the closed unit polydisk \(\overline{\Delta^n}\). The main result of the paper is that then there exists a pluripolar set \(E\) in \(\Delta^k\), independent of \(\alpha\), such that for all \(a\in \Delta^k\setminus E\) the slice \(\langle T,\pi, a\rangle_\alpha\) exists and is in independent of \(\alpha\). For a smooth \((p- k,p-k)\)-form \(\varphi\) in \(\Delta^n\) and a locally bounded plurisubharmonic function \(v\) in \(\Delta^k\) the formula \[ \int_{\Delta^n} T\wedge (dd^c(v\circ \pi))^k\wedge\varphi= \int_{a\in\Delta^k}\langle T,\pi, a\rangle(\varphi)(dd^c v)^k \] gives the connection between the current \(T\) and its slice, and generalizes a formula by Federer (where \(v(z)=|z'|^2\)). The authors also show that for any given pluripolar set \(E\) in \(\Delta^k\) there exists a closed, positive curr…
Mathematics Subject Classification ID: 32U40
Mathematics Subject Classification ID: 32C30
Mathematics Subject Classification ID: 32U25
Mathematics Subject Classification ID: 32W20
zbMATH DE Number: 1542069
zbMATH Keywords: closed positive current
zbMATH Keywords: plurisubharmonic function
zbMATH Keywords: Lelong numbers
zbMATH Keywords: Monge-Ampère operator
zbMATH Keywords: potential
zbMATH Keywords: slicing formula
reviewed by: P-A Mazet
MaRDI profile type: Publication
cites work: Lelong numbers of positive plurisubharmonic currents
cites work: Q3136268
cites work: Q4862177
cites work: Slicing and extension of closed positive currents
cites work: A new capacity for plurisubharmonic functions
cites work: Sur l'existence du cône tangent à un courant positif fermé. (About the existence of the tangent cone with positive closed current)
cites work: Q3310999
cites work: Q3993490
cites work: Q5601113
cites work: Sur le prolongement des courants positifs fermés
cites work: Q4022339
cites work: Q5565773
cites work: Oka's inequality for currents and applications
cites work: A characterization of holomorphic chains
cites work: Extending analytic objects
cites work: Q5590990
cites work: Q4125107
cites work: Q3708270
cites work: Q5512352
cites work: Slicing of closed positive currents and the equation of Lelong-Poincaré
cites work: Über den Flächeninhalt analytischer Mengen udn die Erzeugung k- pseudokonvexer Gebiete
cites work: Q5528072
cites work: Q4081578
cites work: Prolongement des courants, positifs, fermes de masse finie
cites work: Q3728363
cites work: Extension of meromorphic maps into Kähler manifolds
cites work: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents
cites work: Quelques problèmes de prolongement de courants en analyse complexe
cites work: Über die Fortsetzbarkeit analytischer Mengen endlichen Oberflächeninhaltes
description: scientific article; zbMATH DE number 1542069