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Radon-Penrose transform for \({\mathcal D}\)-modules. (English) Zbl 0879.32012
This article is devoted to the proof of the results annonced in [the authors, C. R. Acad. Sci., Paris, Sér. I 319, No. 5, 461-466 (1994; Zbl 0827.32008)]. Let \(X \buildrel{f}\over{\leftarrow} S \buildrel{g}\over{\rightarrow} Y\) be a correspondence of complex analytic manifolds where \(f\) and \(g\) are smooth and \((f,g)\) is a closed embedding. This situation is a generalization of Penrose’s twistor correspondence. For each coherent \({\mathcal D}_X\)-module \({\mathcal M}\) (or more generally for \({\mathcal M}\) in \(D^b_c({\mathcal D}_X)\)) write \({\underline{\Phi}}_S {\mathcal M} = {\underline{g}}_*{\underline{f}}^{-1} {\mathcal M}\) and \({\underline{\Psi}}_{S} {\mathcal M} = {\underline{\Phi}}_S{\mathcal M}[d_Y-d_X]\) where \(d_Z\) is the dimension of the complex manifold \(Z\) and \({\underline{g}}_*\), \({\underline{f}}^{-1}\) are the direct image and the inverse image functor respectively, in the sense of \({\mathcal D}\)-modules. Similarly, for \({\mathcal N}\) in \(D_c^b({\mathcal D}_Y)\) write \({\underline{\Phi}}_{{\widetilde{S}}} {\mathcal N} = {\underline{f}}_*{\underline{g}}^{-1}{\mathcal N}\) and \({\underline{\Psi}}_{{\widetilde{S}}}{\mathcal N} = {\underline{\Phi}}_{{\widetilde{S}}} {\mathcal N} [d_X-d_Y]\). Let write \({\underline{\Phi}}_S^0\) instead of \(H^0\circ{\underline{\Phi}}_S\) and \({\underline{\Psi}}_{{\widetilde{S}}}^0\) instead of \(H^0\circ{\underline{\Psi}}_{{\widetilde{S}}}\) where \(H^0\) denote the \(0\)-th cohomology functor. Assuming the correspondence verifying some geometrical properties, the functor \({\underline{\Phi}}_S^0\) transforms an object of the category \(Mod_{{good}}({\mathcal D}_X;\dot{T}^*X)\) (the localization of the category \(Mod_{{good}}({\mathcal D}_X)\) by the thick subcategory of holomorphic vector bundles endowed with a flat connection) in an object of the category \(Mod_{RS(V)}({\mathcal D}_Y;\dot{T}^*Y)\) (the localization of \(Mod_{RS(V)}({\mathcal D}_Y)\) by the thick subcategory of holomorphic vector bundles endowed with a flat connection). Here \(Mod_{{good}}({\mathcal D}_X)\) denote the smallest subcategory of \(Mod({\mathcal D}_X)\) containing the \({\mathcal D}_X\)-modules which can be endowed with good filtrations on a neighborhood of any compact subset \(X\) and \(Mod_{RS(V)}({\mathcal D}_Y)\) is the thick subcategory of \(Mod_{{good}}({\mathcal D}_Y)\) whose objects have regular singularities on \(V\); \(V\) being a closed regular involutive submanifold of \(\dot{T}^*Y\) defined canonically by the geometry of \((f,g)\).
The main result of the article is: Theorem: The functor \({\underline{\Phi}}_S^0 : Mod_{{good}}({\mathcal D}_X;\dot{T}^*X) \rightarrow Mod_{RS(V)}\) \(({\mathcal D}_Y;\dot{T}^*Y)\) is an equivalence of categories and a quasi-inverse is the functor \({\underline{\Psi}}_{{\widetilde{S}}}^0\).
This result applies in particular to the classical case of Penrose’s twistor correspondence.

32C38 Sheaves of differential operators and their modules, \(D\)-modules
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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