Wedge product of positive currents and balanced manifolds
type: Item
instance of: scholarly article
title: Wedge product of positive currents and balanced manifolds
zbMATH Open document ID: 1143.32020
author: Giovanni Bassanelli
author: Lucia Alessandrini
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2008-05-29T00:00:00Z
review text: The authors study some wedge products of currents on Kählerian and balanced manifolds. Theorem 2.2. Let \(Y\) be a proper analytic subset of a complex manifold \(X\). Let \(S\) be a positive closed \((1,1)\)-current on \(X\), smooth on \(X-Y\), and let \(T\) be a positive pluriharmonic \((k,k)\)-current on \(X\), with \(k+ \dim Y< \dim X.\) Then there exists a unique current on \(X\), denoted by \(S\wedge T\), with the following property: If \(g\) is a solution if \(S=i\partial\overline{\partial}g\) in an open subset \(U\subset X\), and if \(\{g_j\}\) is a sequence of smooth purisubharmonic metrics on \(U\), which converge to \(g\) in \(C^\infty(U-Y)\), then \(\lim_j\partial\overline{\partial}g_j\wedge T=S\wedge T\) in \(U\). Theorem 3.4. If \(M\) is a compact complex manifold of dimension \(n\geq 3\), and \(C\) is an irreducible curve in \(M\) such that \(M-C\) is Kähler, then either \(M\) is Kähler itself or \([C]\) is the component of a boundary and \(M\) is \(p\)-Kähler for every \(p\geq 2\), or \([C]\) is part of the component of a boundary and \(M\) belongs to the Fujiki class \(\mathcal{C}\), i.e., \(M\) is bimeromorphic to a Kähler manifold. In all cases, \(M\) is balanced.
Mathematics Subject Classification ID: 32U40
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 32J17
zbMATH DE Number: 5281226
zbMATH Keywords: wedge product of currents
zbMATH Keywords: positive currents
zbMATH Keywords: plurisubharmonic currents
zbMATH Keywords: Kähler manifolds
zbMATH Keywords: balanced manifolds
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1206734409
OpenAlex ID: W2054350956
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DOI: 10.2748/TMJ/1206734409
description: scientific article; zbMATH DE number 5281226