On a topological counterpart of regularization for holonomic 𝒟-modules
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 27-55.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.140
Classification: 32C38,  14F05
Keywords: Irregular Riemann-Hilbert correspondence, enhanced perverse sheaves, holonomic D-modules
Andrea D’Agnolo 1; Masaki Kashiwara 2

1 Dipartimento di Matematica, Università di Padova via Trieste 63, 35121 Padova, Italy
2 Kyoto University Institute for Advanced study, Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan & Korea Institute for Advanced Study, Seoul 02455, Korea
@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/}
}
TY  - JOUR
TI  - On a topological counterpart of regularization for holonomic $\protect \mathscr{D}$-modules
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2021
DA  - 2021///
SP  - 27
EP  - 55
VL  - 8
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.140/
UR  - https://zbmath.org/?q=an%3A07282221
UR  - https://doi.org/10.5802/jep.140
DO  - 10.5802/jep.140
LA  - en
ID  - JEP_2021__8__27_0
ER  - 
%0 Journal Article
%T On a topological counterpart of regularization for holonomic $\protect \mathscr{D}$-modules
%J Journal de l’École polytechnique — Mathématiques
%D 2021
%P 27-55
%V 8
%I École polytechnique
%U https://doi.org/10.5802/jep.140
%R 10.5802/jep.140
%G en
%F JEP_2021__8__27_0
Andrea D’Agnolo; Masaki Kashiwara. On a topological counterpart of regularization for holonomic $\protect \mathscr{D}$-modules. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 27-55. doi : 10.5802/jep.140. https://jep.centre-mersenne.org/articles/10.5802/jep.140/

[1] A. D’Agnolo & M. Kashiwara - “Riemann-Hilbert correspondence for holonomic D-modules”, Publ. Math. Inst. Hautes Études Sci. 123 (2016), p. 69-197 | Article | MR: 3502097 | Zbl: 1351.32017

[2] A. D’Agnolo & M. Kashiwara - “A microlocal approach to the enhanced Fourier-Sato transform in dimension one”, Adv. Math. 339 (2018), p. 1-59 | Article | MR: 3866893 | Zbl: 1410.32007

[3] A. D’Agnolo & M. Kashiwara - “Enhanced perversities”, J. reine angew. Math. 751 (2019), p. 185-241 | Article | MR: 3956694 | Zbl: 1423.32015

[4] A. D’Agnolo & M. Kashiwara - “Enhanced specialization and microlocalization”, 2019 | 1908.01276

[5] S. Guillermou & P. Schapira - “Microlocal theory of sheaves and Tamarkin’s non displaceability theorem”, in Homological mirror symmetry and tropical geometry, Lect. Notes Unione Mat. Ital., vol. 15, Springer, Cham, 2014, p. 43-85 | Article | MR: 3330785 | Zbl: 1319.32006

[6] Y. Ito & K. Takeuchi - “On irregularities of Fourier transforms of regular holonomic 𝒟-modules”, Adv. Math. 366 (2020), p. 107093, 62 | Article | MR: 4072797 | Zbl: 07183749

[7] M. Kashiwara - “The Riemann-Hilbert problem for holonomic systems”, Publ. RIMS, Kyoto Univ. 20 (1984) no. 2, p. 319-365 | Article | MR: 743382 | Zbl: 0566.32023

[8] M. Kashiwara - “Riemann-Hilbert correspondence for irregular holonomic 𝒟-modules”, Japan. J. Math. 11 (2016) no. 1, p. 113-149 | Article | MR: 3510681 | Zbl: 1351.32001

[9] M. Kashiwara & P. Schapira - Sheaves on manifolds, Grundlehren Math. Wiss., vol. 292, Springer-Verlag, Berlin, Heidelberg, 1990 | MR: 1074006 | Zbl: 0709.18001

[10] M. Kashiwara & P. Schapira - Ind-sheaves, Astérisque, vol. 271, Société Mathématique de France, Paris, 2001 | Numdam | Zbl: 0993.32009

[11] M. Kashiwara & P. Schapira - “Microlocal study of ind-sheaves. I. Micro-support and regularity”, in Autour de l’analyse microlocale, Astérisque, vol. 284, Société Mathématique de France, Paris, 2003, p. 143-164 | Numdam | Zbl: 1053.35009

[12] M. Kashiwara & P. Schapira - Regular and irregular holonomic D-modules, London Math. Soc. Lect. Note Series, vol. 433, Cambridge University Press, Cambridge, 2016 | Article | MR: 3524769 | Zbl: 1354.32008

[13] D. Tamarkin - “Microlocal condition for non-displaceability”, in Algebraic and analytic microlocal analysis, Springer Proc. Math. Stat., vol. 269, Springer, Cham, 2018, p. 99-223 | Article | MR: 3903318 | Zbl: 1416.35019

Cited by Sources: