Let us consider X the complex vector space of square matrices and the associated projective space. Denote the quotient algebra of all -invariant differential operators modulo those vanishing on -invariant functions. We show that the inverse image functor , where is the canonical projection, establishes an equivalence of categories between the category of regular holonomic -modules on the projective space and the quotient category of graded -modules of finite type modulo those supported by . Then we deduce a combinatorial classification of regular holonomic -modules.
Considérons X l'espace vectoriel complexe des matrices carrées et l'espace projectif associé. Notons l'algèbre quotient de tous les opérateurs différentiels -invariants modulo ceux s'annulant sur les fonctions -invariantes. Nous montrons que le foncteur image inverse , où est la projection canonique, établit une équivalence de catégories entre la catégorie des -modules holonômes réguliers sur l'espace projectif et la catégorie quotient des -modules gradués de type fini modulo ceux portés par . On en déduit une classification des -modules holonômes réguliers.
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Philibert Nang 1
@article{CRMATH_2005__340_10_725_0, author = {Philibert Nang}, title = {$ \mathcal{D}$-modules associated to the projective space of $ n\times n$ matrices}, journal = {Comptes Rendus. Math\'ematique}, pages = {725--730}, publisher = {Elsevier}, volume = {340}, number = {10}, year = {2005}, doi = {10.1016/j.crma.2005.04.005}, language = {en}, }
Philibert Nang. $ \mathcal{D}$-modules associated to the projective space of $ n\times n$ matrices. Comptes Rendus. Mathématique, Volume 340 (2005) no. 10, pp. 725-730. doi : 10.1016/j.crma.2005.04.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.04.005/
[1] D-modules holonômes réguliers en une variable, Mathématiques et Physique, Séminaire de L'ENS 1979–1982, Progr. Math., vol. 37, 1983, pp. 313-321
[2] Revue sur le théorie des D-modules et modèles d'opérateurs pseudodifférentiels, Math. Phys. Stud., vol. 12, Kluwer Academic, 1991, pp. 1-31
[3] Perverse sheaves on Grassmannians, Canad. J. Math., Volume 54 (2002) no. 3, pp. 493-532
[4] Perverse sheaves on rank stratifications, Duke Math. J., Volume 96 (1999) no. 2, pp. 317-362
[5] Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France, Volume 85 (1957), pp. 77-99
[6] Algebraic description of certain categories of D-modules, Funktsional. Anal. i Prilozhen., Volume 19 (1985) no. 3, pp. 56-57
[7] On the maximal overdetermined systems of linear partial differential equations I, Publ. Res. Inst. Math. Sci., Volume 10 (1974/1975), pp. 563-579
[8] On holonomic systems of linear partial differential equations II, Invent. Math., Volume 49 (1978), pp. 121-135
[9] Algebraic study of systems of partial differential equations, Mém. Soc. Math. France, Volume 63 (1995) no. 123 (4)
[10] D-modules and microlocal calculus, Iwanami Series in Modern Mathematics, Transl. Math. Monographs, vol. 217, American Mathematical Society, 2003
[11] On holonomic systems of microdifferential equations III: systems with regular singularities, Publ. Res. Inst. Math. Sci., Volume 17 (1981), pp. 813-979
[12] Perverse sheaves with regular singularities along the curve , Comment. Math. Helv., Volume 63 (1988), pp. 89-102
[13] Connexions méromorphes, London Math. Soc. Lecture Note Ser., vol. 201, 1994, pp. 251-261
[14] D-modules associated to the group of similitudes, Publ. Res. Inst. Math. Sci., Volume 35 (1999) no. 2, pp. 223-247
[15] D-modules associated to matrices, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 139-144
[16] D-modules associated to the determinantal singularities, Proc. Japan. Acad. Ser. A, Volume 80 (2004) no. 5, pp. 74-78
[17] L. Narvaez Macarro, Faisceaux pervers dont le support singulier est le germe d'une courble plane irréductible, Thèse de 3ème Cycle, Université Paris VII, 1984
[18] Cycles évanescents et faisceaux pervers. II. Cas des courbes planes réductibles, London Math. Soc. Lecture Note Ser., vol. 201, 1994, pp. 285-323
[19] Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble), Volume 6 (1955–1956), pp. 1-42
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