Andreotti-Norguet correspondence and \(\mathcal D\)-modules
type: Item
instance of: scholarly article
title: Andreotti-Norguet correspondence and \(\mathcal D\)-modules
zbMATH Open document ID: 0972.32006
author: Jean-Louis Frot
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2001-11-09T00:00:00Z
review text: This article uses the theory of \({\mathcal D}\)-modules in integral geometry as developed by \textit{A. D'Agnolo} and \textit{P. Schapira} [J. Funct. Anal. 139, 349-382 (1996; Zbl 0879.32012)] to investigate an integral transform on \({\mathbb{CP}}_n\) due to \textit{A. Andreotti} and \textit{F. Norguet} [Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 21, 31-82 (1967; Zbl 0176.04001)]. More precisely, this transform takes a \(\bar\partial\)-closed \((p,p)\)-form on \({\mathbb{CP}}_n\setminus{\mathbb{CP}}_{n-p-1}\) and integrates it over the \(p\)-dimensional linear cycles to obtain a holomorphic function on the corresponding open subset of \({\text{Gr}}_{p+1}({\mathbb C}^{n+1})\). It is evident that this transform is well-defined on \(H^p({\mathbb{CP}}_n\setminus{\mathbb{CP}}_{n-p-1},\Omega^p)\). The case \(n=3\) and \(p=1\), is familiar in twistor theory where the Grassmannian \({\text{Gr}}_2({\mathbb C}^4)\) is interpreted as compactified complexified Minkowski space. In this case the range of the transform is the kernel of a conformally invariant fourth order differential operator \(P:\Omega^0\to\Omega^4\). It is this result which the author generalises to arbitrary \(n\) and \(k\). In the general case, he explicitly determines a vector-valued fourth order differential operator whose kernel is the range of the Andreotti-Norguet transform. The method of proof, via the \({\mathcal D}\)-module formulae of D'Agnolo and Schapira, is especially interesting.
reviewed by: M. Eastwood
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 32L25
Mathematics Subject Classification ID: 44A12
Mathematics Subject Classification ID: 35A27
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 1437025
zbMATH Keywords: \({\mathcal D}\)-modules
zbMATH Keywords: integral geometry
zbMATH Keywords: cohomology
zbMATH Keywords: correspondence
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195143497
OpenAlex ID: W2064899025
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DOI: 10.2977/PRIMS/1195143497
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description: scientific article; zbMATH DE number 1437025