Riemann-Hilbert correspondence for holonomic D-modules

D’Agnolo, Andrea; Kashiwara, Masaki

doi:10.1007/s10240-015-0076-y

D’Agnolo, Andrea ¹ ; Kashiwara, Masaki ²

¹ Dipartimento di Matematica, Università di Padova via Trieste 63 35121 Padova Italy
² Research Institute for Mathematical Sciences, Kyoto University 606-8502 Kyoto Japan

Publications Mathématiques de l'IHÉS, Tome 123 (2016), pp. 69-197.

Résumé

The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holonomic $D$ -modules and that of constructible sheaves.

In this paper, we prove a Riemann-Hilbert correspondence for holonomic $D$ -modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkin’s work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.

MR Zbl

DOI : 10.1007/s10240-015-0076-y

Mots-clés : Complex Manifold, Full Subcategory, Tensor Category, Natural Morphism, Distinguished Triangle

Affiliations des auteurs :

D’Agnolo, Andrea ¹ ; Kashiwara, Masaki ²

¹ Dipartimento di Matematica, Università di Padova via Trieste 63 35121 Padova Italy
² Research Institute for Mathematical Sciences, Kyoto University 606-8502 Kyoto Japan

@article{PMIHES_2016__123__69_0,
     author = {D{\textquoteright}Agnolo, Andrea and Kashiwara, Masaki},
     title = {Riemann-Hilbert correspondence for holonomic {D-modules}},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {69--197},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {123},
     year = {2016},
     doi = {10.1007/s10240-015-0076-y},
     mrnumber = {3502097},
     zbl = {1351.32017},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-015-0076-y/}
}

TY  - JOUR
AU  - D’Agnolo, Andrea
AU  - Kashiwara, Masaki
TI  - Riemann-Hilbert correspondence for holonomic D-modules
JO  - Publications Mathématiques de l'IHÉS
PY  - 2016
SP  - 69
EP  - 197
VL  - 123
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://archive.numdam.org/articles/10.1007/s10240-015-0076-y/
DO  - 10.1007/s10240-015-0076-y
LA  - en
ID  - PMIHES_2016__123__69_0
ER  -

%0 Journal Article
%A D’Agnolo, Andrea
%A Kashiwara, Masaki
%T Riemann-Hilbert correspondence for holonomic D-modules
%J Publications Mathématiques de l'IHÉS
%D 2016
%P 69-197
%V 123
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U http://archive.numdam.org/articles/10.1007/s10240-015-0076-y/
%R 10.1007/s10240-015-0076-y
%G en
%F PMIHES_2016__123__69_0

D’Agnolo, Andrea; Kashiwara, Masaki. Riemann-Hilbert correspondence for holonomic D-modules. Publications Mathématiques de l'IHÉS, Tome 123 (2016), pp. 69-197. doi : 10.1007/s10240-015-0076-y. http://archive.numdam.org/articles/10.1007/s10240-015-0076-y/

Bibliographie
Cité par

[1.] D. G. Babbitt and V. S. Varadarajan, Local moduli for meromorphic differential equations, Astérisque, 169–170 (1989), 217 pp. | Numdam | MR | Zbl

[2.] D’Agnolo, A. On the Laplace transform for tempered holomorphic functions, Int. Math. Res. Not., Volume 16 (2014), pp. 4587-4623 | DOI | MR | Zbl

[3.] D’Agnolo, A.; Kashiwara, M. On a reconstruction theorem for holonomic systems, Proc. Jpn. Acad., Ser. A, Math. Sci., Volume 88 (2012), pp. 178-183 | DOI | MR | Zbl

[4.] D’Agnolo, A.; Schapira, P. Leray’s quantization of projective duality, Duke Math. J., Volume 84 (1996), pp. 453-496 | DOI | MR | Zbl

[5.] Deligne, P. Équations Différentielles à Points Singuliers Réguliers (1970) (iii + 133 pp.) | DOI | MR | Zbl

[6.] Deligne, P.; Malgrange, B.; Ramis, J.-P. Singularités Irrégulières, Correspondance et documents (2007) (xii + 188 pp.) | MR | Zbl

[7.] Guillermou, S.; Schapira, P. Microlocal theory of sheaves and Tamarkin’s non displaceability theorem, Homological Mirror Symmetry and Tropical Geometry (2014), pp. 43-85 | DOI | MR | Zbl

[8.] Hille, E. Ordinary Differential Equations in the Complex Domain (1976) (xi + 484 pp.) | MR | Zbl

[9.] Kashiwara, M. The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., Volume 20 (1984), pp. 319-365 | DOI | MR | Zbl

[10.] Kashiwara, M. $D$ -modules and Microlocal Calculus (2003) (xvi + 254 pp.) | MR | Zbl

[11.] Kashiwara, M.; Schapira, P. Sheaves on Manifolds (1990) (x + 512 pp.) | DOI | MR | Zbl

[12.] M. Kashiwara and P. Schapira, Moderate and formal cohomology associated with constructible sheaves, Mém. Soc. Math. France, 64 (1996), iv + 76 pp. | Numdam | MR | Zbl

[13.] M. Kashiwara and P. Schapira, Ind-sheaves, Astérisque, 271 (2001), 136 pp. | Numdam | MR | Zbl

[14.] Kashiwara, M.; Schapira, P. Microlocal study of ind-sheaves. I. Micro-support and regularity, Astérisque, Volume 284 (2003), pp. 143-164 | Numdam | MR | Zbl

[15.] Kashiwara, M.; Schapira, P. Categories and Sheaves (2006) (x + 497 pp.) | DOI | Zbl

[16.] Kedlaya, K. S. Good formal structures for flat meromorphic connections, I: surfaces, Duke Math. J., Volume 154 (2010), pp. 343-418 | DOI | MR | Zbl

[17.] Kedlaya, K. S. Good formal structures for flat meromorphic connections, II: excellent schemes, J. Am. Math. Soc., Volume 24 (2011), pp. 183-229 | DOI | MR | Zbl

[18.] Majima, H. Asymptotic Analysis for Integrable Connections with Irregular Singular Points (1984) (xiv + 249 pp.) | DOI | Zbl

[19.] Mochizuki, T. Good formal structure for meromorphic flat connections on smooth projective surfaces, Algebraic Analysis and Around (2009), pp. 223-253 | Zbl

[20.] T. Mochizuki, Wild harmonic bundles and wild pure twistor $D$ -modules, Astérisque, 340, (2011), x + 607 pp. | Numdam | Zbl

[21.] Morando, G. An existence theorem for tempered solutions of $D$ -modules on complex curves, Publ. Res. Inst. Math. Sci., Volume 43 (2007), pp. 625-659 | DOI | MR | Zbl

[22.] Morando, G. Preconstructibility of tempered solutions of holonomic $D$ -modules, Int. Math. Res. Not., Volume 4 (2014), pp. 1125-1151 | DOI | MR | Zbl

[23.] C. Sabbah, Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque, 263, (2000), viii + 190 pp. | Numdam | Zbl

[24.] Sabbah, C. Introduction to Stokes Structures (2013) (xiv + 249 pp.) | DOI | Zbl

[25.] D. Tamarkin, Microlocal condition for non-displaceability, 2008, 93 pp., | arXiv

[26.] Wasow, W. Asymptotic Expansions for Ordinary Differential Equations (1965) (ix + 362 pp.) | Zbl

Cité par Sources :