Leray's quantization of projective duality
Let \({\mathbf P}\) be a complex \(n\)-dimensional projective space, \({\mathbf P}^*\) its dual space and \({\mathbf A}:=\{(z,\zeta)\in {\mathbf P}\times{\mathbf P}^* |z\in\zeta\}\). The symbol \({\mathcal D}_{{\mathbf P}}\) denote the sheaf of linear differential operators on \({\mathbf P}\). For \(k\in{\mathbf Z}\) denote by \({\mathcal D}_{\mathbf P}(k)\) the sheaf \({\mathcal D}_{\mathbf P}\otimes_{{\mathcal O}_{\mathbf P}}{\mathcal O}_{\mathbf P}(k)\). For each coherent \({\mathcal D}_{\mathbf P}\)-module \({\mathcal M}\) let define \({\underline{\Phi}}_{{\mathbf A}}{\mathcal M} = {\underline{g}}_*{\underline{f}}^{-1} {\mathcal M}\) where \({\mathbf P} \buildrel{f}\over{\leftarrow} {\mathbf A} \buildrel{g}\over{\rightarrow} {\mathbf P}^*\) is the correspondence defined by the projections and \({\underline{g}}_*, {\underline{f}}^{-1}\) are the direct image and the inverse image functors respectively, in the sense of \({\mathcal D}\)-modules. The main result of the article is that for \(-n-1<k<0\) there is a natural isomorphism \({\mathcal D}_{{\mathbf P}^*}(-k^*) \rightarrow {\underline{\Phi}}_{\mathbf A}({\mathcal D}_{\mathbf P}(k))\) where \(k^* = -n-1-k\). The paper contains two proofs of the main result. The first one uses globally defined contact transformations and the second one uses kernels of sheaves generalising classical results of Leray about differential and integral calculus on complex varieties. The proofs use previous results of the authors on general correspondences of complex varieties. As applications they generalize a theorem of Martineau [\textit{A. Martineau}, C. R. Acad. Sci., Paris Sér. I 255, 2888-2890; (1962; Zbl 0195.36603)] and give an alternative approach to the results of Gelfand, Gindikin and Graev on real Radon transform [\textit{I. M. Gelfand, S.G. Gindikin} and \textit{M. I. Graev}, J. Sov. Math. 18, 39-167 (1982; Zbl 0474.52010)].
scientific article; zbMATH DE number 926954
scientific article