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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.

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