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Radon and Fourier transforms for $$\mathcal D$$-modules. (English) Zbl 1051.32008
The authors generalize the result of J.-L. Brylinski [Astérisque 140–141, 3–134 (1986; Zbl 0624.32009)], in order to consider more general Radon transforms, to treat quasi-coherent $$\mathcal{D}$$-modules and twisted sheaves. First, they give a new proof of the fact that the natural morphism $$\mathcal{D}_{\mathbb{P}} (-m^*) \circ {}^{\mathcal{D}}\mathcal{R} \mathcal{D} \det \mathbb{V}\to \mathcal{D}_{p^* (-m)}$$ is an isomorphism, proved by Brylinski (loc. cit.). Here $$\mathbb{V}$$ is a vector space, $$\mathbb{P} = \mathbb{P}(\mathbb{V})$$ is the projective space of lines in $$\mathbb{V}$$, $$(\circ)_{\cdot}^{\mathbb{D}} \mathcal{R}$$ denotes the Radon transform associated with $$U \subset \mathbb{P} \times \mathbb{P}^*$$ ($$U =$$ the complement of $$\mathbb{S}$$ in $$\mathbb{P} \times \mathbb{P}^*$$, the hypersurface of pairs $$(x, y)$$ where $$x$$ is a point belonging to the hyperplane $$y \in \mathbb{P}^*$$), $$m \in \mathbb{Z}$$, $$m^* = -m - n - 1$$, $$\mathcal{D}_{\mathbb{P}} (m) = \mathcal{D}_{\mathbb{P}} \otimes_{\mathcal{O}} (m)$$ where $$\mathcal{O}_{\mathbb{P}} (\mathbb{P}^m)$$ is the $$m$$-th tensor power of the topological line bundle $$\mathcal{O}_{\mathbb{P}}(-1)$$ and $$n$$ is the dimension of $$\mathbb{P}$$. In fact, the authors prove more: they obtain a description of the Radon transform of $$\mathcal{D}_{\mathbb{P}}(-m^*)$$ for $$\geq 0$$, namely they show that there is a long exact sequence of $$\mathcal{D}_{\mathbb{P}^*}$$-modules $0 \to \mathcal{O}_{\mathbb{P}^*} \otimes S^m \mathbb{V}^* \to \mathcal{D}_p (-m^*) \circ {}^{\mathbb{D}} \mathcal{D} \otimes \operatorname {der} \mathbb{V} \to \mathcal{D}_{\mathbb{P}^*}(-n) \to \mathcal{O}_{\mathbb{P}^*} \otimes S^m \mathbb{V}^*$ (here $$S^m \mathbb{V}$$ is the $$m$$-th symmetric tensor of $$\mathbb{V}$$). For differential forms, let $$\mathcal{S}_{p.}^{\mathbb{D}}$$ be the Spencer complex; the shifted complex $$\mathcal{S}_{p \geq q}^{\mathbb{P}}[q]$$ describes the sheaf of closed $$q$$-forms. Then (theorem 7) one has the isomorphism $$\mathcal{S}_p^{\mathbb{P}} [q]^{\mathbb{P}} \circ \mathcal{R}\mathop{\sim}\limits_{\leftarrow} \mathcal{S}p^{\mathbb{P}^*}_{p\geq n - q} [n - q]$$. This result is obtained by relating (theorem 8) the Radon transform of the sheaf of $$q$$-forms with the subsheaf of $$j^{-1} \Omega_{\mathbb{V}^*}^{n + 1 - q}$$ of sections $$\sigma$$ satisfying $$l_\theta \sigma = d \sigma = 0$$ where $$j : \mathbb{V}\backslash \{0\} @>j>> \mathbb{V}$$ and $$\theta$$ is the Euler vector field.
The proofs are entirely geometric and consist mainly of a reduction to the one-dimensional case, by the use homogeneous coordinates. Finally the authors give a quantization [in the sense of A. D’Agnolo and P. Schapira, Duke Math. J. 84, No. 2, 453–496 (1996; Zbl 0879.32011)] of the Radon transform for differential forms.

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules 58J10 Differential complexes
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##### References:
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