Grassmann duality for \({\mathcal D}\)-modules. (La dualité de Grassmann pour les \({\mathcal D}\)-modules.) (French. Abridged English version) Zbl 0860.32005

Author’s summary: “Let \(\mathbb{G}\) and \(\mathbb{G}^*\) be two Grassmann manifolds of subspaces of complementary dimensions in a complex vector space, and let \(\Omega\) be the open subset of transversal pairs in \(\mathbb{G} \times \mathbb{G}^*\). Generalizing results of J.-L. Brylinski [Astérisque 140-141, 3-134 (1986; Zbl 0624.32009)], A. D’Agnolo and P. Schapira [C. R. Acad. Sci., Paris, Sér. I 319, No. 6, 595-598 (1994; Zbl 0832.32013)] and M. Kashiwara and T. Tanisaki [Duke Math. J. 77, No. 1, 21-62 (1995; Zbl 0829.17020)] for projective duality, we show that the integral transform defined by the constant sheaf on \(\Omega\) is an equivalence between the bounded derived categories of sheaves on \(\mathbb{G}\) and \(\mathbb{G}^*\) with \(\mathbb{R}\)- or \(\mathbb{C}\)-constructible cohomology. Moreover the integral transform given by the associated regular holonomic kernel is an equivalence between bounded derived categories of \(\mathcal D\)-modules on \(\mathbb{G}\) and \(\mathbb{G}\) with coherent or regular holonomic cohomology”.


32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials