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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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