The \(\overline\partial\)-equation on a positive current
type: Item
instance of: scholarly article
title: The \(\overline\partial\)-equation on a positive current
zbMATH Open document ID: 1031.32005
author: Bo Berndtsson
author: Nessim Sibony
published in: Inventiones Mathematicae
publication date: 2002-08-08T00:00:00Z
review text: The authors study the induced \(\overline\partial\)-equation on a positive current in a complex manifold \(M\). More precisely, let \(u\) and \(f\) be smooth differential forms, \(T\) a positive current on \(M\). \(\overline\partial u=f\) on \(T\) means -- by definition -- that (*) \(\overline\partial u\wedge T=f\wedge T\). In the case when \(T=[V]\), the current of integration on an analytic set \(V\) in \(M\), (*) means that \(\overline \partial(u |_{V_{\text{reg}}})= f|_{V_{\text{reg}}}\) where \(V_{\text{reg}}\) denote the set of regular points of \(V\). The authors obtain a number of very interesting results, mostly for currents of bidimension (1,1). Let \(T\) be a positive current of bidimension \((p,p),\omega\) a smooth (1,1) form, \(\sigma_T=T \wedge\omega_p\) where \(\omega_p= {\omega^p \over p!}\) the trace measure of \(T\). For \(g\) a \((0,q)\) form in \(\mathbb{C}^n\), the \(L^2\)-norm of \(g\) over \(T\) is \(\|g\|^2= \int c_q g\wedge\overline g\wedge T\wedge \omega_{p-q}\) if \(T\) is of bidimension \((p,p)\); here \(c_q=(-1)^{q (q+1)/2} i^q=(-i)^{q^2}\) ensure that the \(c_qg \wedge \overline g\) is a positive form. If \(f\) is a \((p,q)\) form, define \(\|f\|=\sup |\int f\wedge g\wedge T|\) the supremum being taken over all \(g(0,p-q)\) forms, with \(\|g\|<1\). Let \(L^2_{(p,q)}(T)\) be the completion of the space of \((p,q)\) forms with respect to the norms defined above. Another definition is that of \(\Gamma\)-directed currents of bidimension \((p,p)\). For \(M\) a complex manifold with a Hermitian metric \(g\) and \(\omega\) the associated \((1,1)\) form, for \(X\subset M\) assume that for every \(z\in X\) then is given a cone \(\Gamma_z\) of positive currents of bidimension \((p,p)\), \(\varepsilon_Z\) supported at \(z\). Let \(\Gamma=\cup_{z\in X\gamma_z}\), assumed to be closed. Then \(T\) is \(\Gamma\)-directed if \(T\) belongs to the convex hull of \(\Gamma\); \(T\) is said to be \(\alpha\)-directed when \(\Gamma\) is defined by the Dirac c…
reviewed by: Gheorghe Gussi
Mathematics Subject Classification ID: 32C30
Mathematics Subject Classification ID: 32W05
zbMATH DE Number: 1777228
zbMATH Keywords: positive currents
zbMATH Keywords: existence for the \(\overline\partial\)-equation
zbMATH Keywords: estimates of solutions
MaRDI profile type: Publication
full work available at URL: s002220100178
OpenAlex ID: W2091403111
DOI: 10.1007/S002220100178
description: scientific article; zbMATH DE number 1777228