Leray’s quantization of projective duality. (English) Zbl 0879.32011

Let \({\mathbf P}\) be a complex \(n\)-dimensional projective space, \({\mathbf P}^*\) its dual space and \({\mathbf A}:=\{(z,\zeta)\in {\mathbf P}\times{\mathbf P}^* |z\in\zeta\}\). The symbol \({\mathcal D}_{{\mathbf P}}\) denote the sheaf of linear differential operators on \({\mathbf P}\). For \(k\in{\mathbf Z}\) denote by \({\mathcal D}_{\mathbf P}(k)\) the sheaf \({\mathcal D}_{\mathbf P}\otimes_{{\mathcal O}_{\mathbf P}}{\mathcal O}_{\mathbf P}(k)\). For each coherent \({\mathcal D}_{\mathbf P}\)-module \({\mathcal M}\) let define \({\underline{\Phi}}_{{\mathbf A}}{\mathcal M} = {\underline{g}}_*{\underline{f}}^{-1} {\mathcal M}\) where \({\mathbf P} \buildrel{f}\over{\leftarrow} {\mathbf A} \buildrel{g}\over{\rightarrow} {\mathbf P}^*\) is the correspondence defined by the projections and \({\underline{g}}_*, {\underline{f}}^{-1}\) are the direct image and the inverse image functors respectively, in the sense of \({\mathcal D}\)-modules.
The main result of the article is that for \(-n-1<k<0\) there is a natural isomorphism \({\mathcal D}_{{\mathbf P}^*}(-k^*) \rightarrow {\underline{\Phi}}_{\mathbf A}({\mathcal D}_{\mathbf P}(k))\) where \(k^* = -n-1-k\).
The paper contains two proofs of the main result. The first one uses globally defined contact transformations and the second one uses kernels of sheaves generalising classical results of Leray about differential and integral calculus on complex varieties. The proofs use previous results of the authors on general correspondences of complex varieties. As applications they generalize a theorem of Martineau [A. Martineau, C. R. Acad. Sci., Paris Sér. I 255, 2888-2890; (1962; Zbl 0195.36603)] and give an alternative approach to the results of Gelfand, Gindikin and Graev on real Radon transform [I. M. Gelfand, S.G. Gindikin and M. I. Graev, J. Sov. Math. 18, 39-167 (1982; Zbl 0474.52010)].


32C38 Sheaves of differential operators and their modules, \(D\)-modules
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[1] L. A. Aĭ zenberg, Linear convexity in C\(^{n}\) and the distribution of the singularities of holomorphic functions , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 487-495. · Zbl 0159.37402
[2] E. Andronikof, Microlocalisation tempérée , Mém. Soc. Math. France (N.S.) 122 (1994), no. 57, 176. · Zbl 0805.58059
[3] R. J. Baston and M. G. Eastwood, The Penrose transform , Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1989. · Zbl 0726.58004
[4] A. Beĭ linson, J. N. Bernstein, and P. Deligne, Faisceaux pervers , Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5-171. · Zbl 0536.14011
[5] J. L. Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques , Astérisque (1986), no. 140-141, 3-134, 251. · Zbl 0624.32009
[6] 1 A. D’Agnolo and P. Schapira, La transformée de Radon-Penrose des \({\scr D}\)-modules , C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 5, 461-466, paper to appear in J. Funct. Anal. 138, 1996. · Zbl 0827.32008
[7] 2 A. D’Agnolo and P. Schapira, Correspondence for \({\scr D}\)-modules and Penrose transform , Topol. Methods Nonlinear Anal. 3 (1994), no. 1, 55-68. · Zbl 0809.32003
[8] A. D’Agnolo and P. Schapira, Quantification de Leray de la dualité projective , C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 6, 595-598. · Zbl 0832.32013
[9] C. Ehresmann, Sur la topologie de certaines variétées algébriques réelles , Journ. de Math. 16 (1937), 69-100. · Zbl 0016.07403
[10] I. M. Gelfand, S. G. Gindikin, and M. I. Graev, Integral geometry in affine and projective spaces , J. Soviet Math. 18 (1982), 39-167. · Zbl 0474.52010
[11] S. G. Gindikin and G. M. Khenkin, Integral geometry for \(\overline\partial\)-cohomology in \(q\)-linear concave domains in \(\mathbb{CP}^n\) , Functional Anal. Appl. 12 (1979), 247-261. · Zbl 0423.32013
[12] G. M. Henkin, The Abel-Radon transform and several complex variables , preprint, 1993. · Zbl 0848.32012
[13] G. M. Henkin and J. Leiterer, Andreotti-Grauert theory by integral formulas , Progress in Mathematics, vol. 74, Birkhäuser Boston Inc., Boston, MA, 1988. · Zbl 0654.32002
[14] M. Kashiwara, Systems of microdifferential equations , Progress in Mathematics, vol. 34, Birkhäuser Boston Inc., Boston, MA, 1983. · Zbl 0521.58057
[15] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems , Publ. Res. Inst. Math. Sci. 20 (1984), no. 2, 319-365. · Zbl 0566.32023
[16] M. Kashiwara and T. Kawai, On holonomic systems of microdifferential equations. III. Systems with regular singularities , Publ. Res. Inst. Math. Sci. 17 (1981), no. 3, 813-979. · Zbl 0505.58033
[17] M. Kashiwara, T. Kawai, and T. Kimura, Foundations of algebraic analysis , Princeton Mathematical Series, vol. 37, Princeton Univ. Press, Princeton, NJ, 1986. · Zbl 0605.35001
[18] M. Kashiwara and P. Schapira, Microlocal study of sheaves , Astérisque (1985), no. 128, 235. · Zbl 0589.32019
[19] M. Kashiwara and P. Schapira, Sheaves on manifolds , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. · Zbl 0709.18001
[20] M. Kashiwara and P. Schapira, Moderate and formal cohomology associated with constructible sheaves , preprint 999, Res. Inst. Math. Sci. Kyoto Univ., 1994, and paper · Zbl 0881.58060
[21] M. Kashiwara and T. Tanisaki, Kazhdan-Lusztig conjecture for affine Lie algebras with negative level II. Non-integral case , preprint 1029, Res. Inst. Math. Sci. Kyoto Univ., 1995, and article to appear in Duke Math. J. · Zbl 0829.17020
[22] J. Leray, Le calcul différentiel et intégral sur une variété analytique complexe. (Problème de Cauchy. III) , Bull. Soc. Math. France 87 (1959), 81-180. · Zbl 0199.41203
[23] A. Martineau, Indicatrices des fonctions analytiques et inversion de la transformation de Fourier-Borel par la transformation de Laplace , C. R. Acad. Sci. Paris 255 (1962), 2888-2890. · Zbl 0195.36603
[24] A. Martineau, Equations différentielles d’ordre infini , Bull. Soc. Math. France 95 (1967), 109-154. · Zbl 0167.44202
[25] M. Sato, T. Kawai, and M. Kashiwara, Microfunctions and pseudo-differential equations , Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Springer, Berlin, 1973, 265-529. Lecture Notes in Math., Vol. 287. · Zbl 0277.46039
[26] P. Schapira, Microdifferential systems in the complex domain , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 269, Springer-Verlag, Berlin, 1985. · Zbl 0554.32022
[27] P. Schapira and J.-P. Schneiders, Elliptic pairs. II. Euler class and relative index theorem , Astérisque (1994), no. 224, 61-98. · Zbl 0856.58039
[28] J.-P. Schneiders, An introduction to \(\scr D\)-modules , Bull. Soc. Roy. Sci. Liège 63 (1994), no. 3-4, 223-295. · Zbl 0816.35004
[29] B. Sternin and V. Shatalov, Differential equations on complex manifolds , Mathematics and its Applications, vol. 276, Kluwer Academic Publishers Group, Dordrecht, 1994. · Zbl 0818.35003
[30] J. M. Trépreau, Transformation de Legendre et pseudoconvexité avec décalage , J. Fourier Anal. Appl. (1995), no. Special Issue, 569-588, Kahane Special Issue. · Zbl 0891.32009
[31] S. V. Znamenskiĭ, A geometric criterion of strong linear convexity , Funktsional. Anal. i Prilozhen. 13 (1979), no. 3, 83-84. · Zbl 0427.32018
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