Andreotti-Norguet correspondence and \(\mathcal D\)-modules
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.
scientific article; zbMATH DE number 1437025
scientific article