On a topological counterpart of regularization for holonomic 𝒟-modules
[Sur un analogue topologique de la régularisation pour les 𝒟-modules holonomes]
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 27-55.

Sur une variété complexe lisse, l’inclusion de la catégorie des 𝒟-modules holonomes réguliers dans celle des 𝒟-modules holonomes admet un foncteur quasi-inverse à gauche reg , appelé régularisation. Rappelons que reg est reconstruit à partir du complexe de de Rham de par la correspondance de Riemann-Hilbert régulière. De même, sur un espace topologique, l’inclusion des faisceaux dans les ind-faisceaux enrichis admet un foncteur quasi-inverse à gauche, qu’on appelle ici faisceautisation. La régularisation et la faisceautisation sont échangées par la correspondance de Riemann-Hilbert irrégulière. Dans ce travail, nous étudions certaines des propriétés du foncteur de faisceautisation. En particulier, nous fournissons une formule qui calcule la fibre du faisceautisé de la spécialisation et de la microlocalisation enrichies.

On a complex manifold, the embedding of the category of regular holonomic 𝒟-modules into that of holonomic 𝒟-modules has a left quasi-inverse functor reg , called regularization. Recall that reg is reconstructed from the de Rham complex of by the regular Riemann-Hilbert correspondence. Similarly, on a topological space, the embedding of sheaves into enhanced ind-sheaves has a left quasi-inverse functor, called here sheafification. Regularization and sheafification are intertwined by the irregular Riemann-Hilbert correspondence. Here, we study some of the properties of the sheafification functor. In particular, we provide a stalk formula for the sheafification of enhanced specialization and microlocalization.

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DOI : https://doi.org/10.5802/jep.140
Classification : 32C38,  14F05
Mots clés : Correspondance de Riemann-Hilbert irrégulière, faisceau pervers enrichi, D-module holonome
@article{JEP_2021__8__27_0,
     author = {Andrea D{\textquoteright}Agnolo and Masaki Kashiwara},
     title = {On a topological counterpart of regularization for holonomic $\protect \mathscr{D}$-modules},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {27--55},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.140},
     zbl = {07282221},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.140/}
}
Andrea D’Agnolo; Masaki Kashiwara. On a topological counterpart of regularization for holonomic $\protect \mathscr{D}$-modules. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 27-55. doi : 10.5802/jep.140. https://jep.centre-mersenne.org/articles/10.5802/jep.140/

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