A note on polynomial convexity: Poletsky disks, Jensen measures and positive currents
type: Item
instance of: scholarly article
title: A note on polynomial convexity: Poletsky disks, Jensen measures and positive currents
zbMATH Open document ID: 1226.32006
author: Erlend Fornæss Wold
published in: The Journal of Geometric Analysis
publication date: 2011-05-13T00:00:00Z
review text: \textit{J. Duval} and \textit{N. Sibony} [Duke Math. J. 79, No. 2, 487--513 (1995; Zbl 0838.32006)] proved that a point \(x\) belongs to the polynomial hull \(\widehat{K}\) of a compact set \(K\) in \(\mathbb C^n\) if and only if there exists a positive \((1,1)\)-current \(T\) with \(x\in\text{supp\,}T\) such that \(dd^cT=\widetilde{\sigma}-\delta_x\), where \(\widetilde{\sigma}\) is a representative Jensen measure for (evaluation at) \(x\). Earlier, \textit{E. A. Poletsky} [Indiana Univ. Math. J. 42, No. 1, 85--144 (1993; Zbl 0811.32010)] showed that if \(K\) is a compact subset of a bounded pseudoconvex Runge domain \(\Omega\) in \(\mathbb C^n\), then \(x\in\widehat{K}\) if and only if the following holds: for any open neighborhood \(U\) of \(K\) and for any \(\varepsilon>0\), there exists a holomorphic map \(f: \overline\Delta\to\Omega\) such that \(f(0)=x\) and \[ \sigma\Big(\big\{\zeta\in\partial\Delta\;:\;f(\zeta)\notin U\big\}\Big)<\varepsilon. \] Here \(\sigma\) denotes the normalized Lebesgue measure on the boundary of the unit disk \(\Delta\) in \(\mathbb C\). In this paper, the author shows that the currents \(T\) of Duval and Sibony can be (relatively easily) obtained as weak limits of push-forwards of the Green current \[ G(\omega):=-\iint_{\Delta}\log|\zeta|\cdot\omega,\qquad\omega\in\mathcal E^{1.1}\big(\overline\Delta\big), \] by the Poletsky maps.
reviewed by: Wiesław Pleśniak
Mathematics Subject Classification ID: 32E20
zbMATH DE Number: 5887812
zbMATH Keywords: polynomial convexity
zbMATH Keywords: positive currents, Poletsky disks, Jensen measures
MaRDI profile type: Publication
full work available at URL: s12220-010-9130-7
OpenAlex ID: W2054036616
cites work: Polynomial convexity, rational convexity, and currents
cites work: Q4298658
cites work: Q5720347
DOI: 10.1007/S12220-010-9130-7
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description: scientific article; zbMATH DE number 5887812