Coleff-Herrera currents, duality, and noetherian operators
type: Item
instance of: scholarly article
title: Coleff-Herrera currents, duality, and noetherian operators
zbMATH Open document ID: 1241.32007
author: Mats Andersson
publication date: 2012-04-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/0902.3064
full work available at URL: smf bull 139 535-554.php
Mathematics Subject Classification ID: 32C30
Mathematics Subject Classification ID: 32A27
Mathematics Subject Classification ID: 32C15
zbMATH DE Number: 6024214
zbMATH Keywords: Coleff-Herrera current
zbMATH Keywords: duality
zbMATH Keywords: Noetherian operators
zbMATH Keywords: residue currents
MaRDI profile type: Publication
OpenAlex ID: W1530193761
arXiv ID: 0902.3064
cites work: Integral representation with weights. II: Division and interpolation
cites work: Uniqueness and factorization of Coleff-Herrera currents
cites work: Residue currents with prescribed annihilator ideals
cites work: Decomposition of residue currents
cites work: Q3964695
cites work: Q4669634
cites work: Q5505184
cites work: Les courants residuels associes à une forme méromorphe
cites work: Canonical representatives in moderate cohomology
cites work: Q5590963
cites work: Q4317713
cites work: Q3215513
cites work: A local Grothendieck duality for Cohen-Macaulay ideals
cites work: The construction of Noetherian operators
cites work: Q5585729
cites work: Residues, currents, and their relation to ideals of holomorphic functions.
cites work: Q3846409
DOI: 10.24033/BSMF.2618
published in: Bulletin de la Société Mathématique de France
review text: Let \(f=(f_1, \dots, f_p)\) be a tuple of holomorphic functions in a neighborhood of the origin in \(\mathbb{C}^n\) that defines a complete intersection. Then the Coleff-Herrera product NEWLINE\[NEWLINE\mu^f = \overline{\partial} \frac{1}{f_1} \wedge \cdots \wedge \overline{\partial} \frac{1}{f_p},NEWLINE\]NEWLINE introduced in [\textit{N. R. Coleff} and \textit{M. E. Herrera}. Les courants residuels associes à une forme méromorphe, Lect. Notes Math. 633. Berlin etc.: Springer (1978; Zbl 0371.32007)] is a \(\overline{\partial}\)-closed \((0,p)\)-current with support on \(\{f=0\}\), and it is independent (up to a non-vanishing holomorphic factor) of the choice of generators of the ideal sheaf \(\mathcal{I}\) generated by \(f\). It was proved in [\textit{A. Dickenstein} and \textit{C. Sessa}, Invent. Math. 80, 417--434 (1985; Zbl 0556.32005)] and [\textit{U. Oberst}, J. Algebra 222, No. 2, 595--620 (1999; Zbl 0948.13015)] that \(\mathcal{I}\) coincides with the ideal sheaf \(\text{ann\,} \mu^f\) of holomorphic functions \(\phi\) such that the current \(\mu^f\phi\) vanishes. This is often referred to as the duality principle. NEWLINENEWLINENEWLINE The Coleff-Herrera current described above is the model for the general notion of Coleff-Herrera currents introduced by Björk: given a variety \(Z\) of pure codimension \(p\), we say that a (possibly vector valued) \((0,p)\)-current \(\mu\) with support on \(Z\) is a Coleff-Herrera current on \(Z\) if it is \(\overline{\partial}\)-closed, annihilated by \(\overline{\mathcal{I}_Z}\) (i.e. \(\overline{\xi}\mu=0\) for each holomorphic function \(\xi\) vanishing on \(Z\)), and has the standard extension property SEP (see [\textit{J.-E. Björk}, Residues and \(\mathcal{D}\)-modules. in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, Oslo, Norway, 2002. Berlin: Springer. 605--651 (2004; Zbl 1069.32001)] or [the author, Ann. Fac. Sci. Toulouse, Math. (6) 18, No. 4, 651--661 (2009; Zbl 1187.32026)]). NE…
reviewed by: Jean Ruppenthal
description: scientific article; zbMATH DE number 6024214