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Riemann-Hilbert correspondence for holonomic \(\mathcal{D}\)-modules. (English) Zbl 1351.32017
Let \(X\) be a complex manifold. The classical Riemann-Hilbert correspondence in a modern form [M. Kashiwara, Publ. Res. Inst. Math. Sci. 20, 319–365 (1984; Zbl 0566.32023)] associates to a regular holonomic system the \(\mathbb C\)-constructible complex of its holomorphic solutions. In particular, the corresponding \({\mathcal D}_X\)-module \(\mathcal L\) can be reconstructed from the holomorphic de Rham complex of \(\mathcal L\) denoted by \(\Omega_X\otimes_{{\mathcal D}_X}^L\!\!{\mathcal L}\), where \(\Omega_X\) is the sheaf of holomorphic differential forms of highest degree. In their previous work [Proc. Japan Acad., Ser. A 88, No. 10, 178–183 (2012; Zbl 1266.32012)], the authors extended this correspondence to the case of irregular holonomic \({\mathcal D}_X\)-modules given on a complex curve. More precisely, they constructed a contravariant functor from the derived category of \({\mathcal D}_X\)-modules to the derived category of ind-sheaves on \(X\times \mathbb P^1_{\mathbb C}\), so that any irregular holonomic \({\mathcal D}_X\)-module \(\mathcal M\) can be reconstructed from an object of the target, the so-called tempered de Rham complex of \(\mathcal M\) associated with the complex of tempered holomorphic functions on \(X\) (see [M. Kashiwara and P. Schapira, in: Autour de l’analyse microlocale. Paris: Société Mathématique de France. 143–164 (2003; Zbl 1053.35009)]). In the paper under review, slightly modifying the target category, the authors describe in detail a similar construction for manifolds of higher dimensions. They also underline that their approach was inspired by [D. Tamarkin, Microlocal condition for non-displaceablility, arXiv:0809.1584] and is mainly based on the theory of ind-sheaves by M. Kashiwara and P. Schapira [Sheaves on manifolds. With a short history “Les débuts de la théorie des faisceaux” by Christian Houzel. Berlin etc.: Springer-Verlag (1990; Zbl 0709.18001)], the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya, etc.

32C38 Sheaves of differential operators and their modules, \(D\)-modules
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
35Q15 Riemann-Hilbert problems in context of PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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[1] D. G. Babbitt and V. S. Varadarajan, Local moduli for meromorphic differential equations, Astérisque, 169-170 (1989), 217 pp. · Zbl 0683.34003
[2] D’Agnolo, A., On the Laplace transform for tempered holomorphic functions, Int. Math. Res. Not., 16, 4587-4623, (2014) · Zbl 1304.32006
[3] D’Agnolo, A.; Kashiwara, M., On a reconstruction theorem for holonomic systems, Proc. Jpn. Acad., Ser. A, Math. Sci., 88, 178-183, (2012) · Zbl 1266.32012
[4] D’Agnolo, A.; Schapira, P., Leray’s quantization of projective duality, Duke Math. J., 84, 453-496, (1996) · Zbl 0879.32011
[5] P. Deligne, Équations Différentielles à Points Singuliers Réguliers, Lecture Notes in Mathematics, vol. 163, Springer, Berlin, 1970, iii + 133 pp. · Zbl 0244.14004
[6] P. Deligne, B. Malgrange and J.-P. Ramis, Singularités Irrégulières, Correspondance et documents, Documents Mathématiques, vol. 5, Société Mathématique de France, Paris, 2007, xii + 188 pp. · Zbl 1130.14001
[7] Guillermou, S.; Schapira, P., Microlocal theory of sheaves and tamarkin’s non displaceability theorem, No. 15, 43-85, (2014), Berlin · Zbl 1319.32006
[8] E. Hille, Ordinary Differential Equations in the Complex Domain, Pure and Applied Mathematics, Wiley-Interscience, New York, 1976, xi + 484 pp. · Zbl 0343.34007
[9] Kashiwara, M., The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., 20, 319-365, (1984) · Zbl 0566.32023
[10] M. Kashiwara, \(\mathcal{D}\)-modules and Microlocal Calculus, Translations of Mathematical Monographs, vol. 217, Am. Math. Soc., Providence, 2003, xvi + 254 pp. · Zbl 1017.32012
[11] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292, Springer, Berlin, 1990, x + 512 pp. · Zbl 0709.18001
[12] M. Kashiwara and P. Schapira, Moderate and formal cohomology associated with constructible sheaves, Mém. Soc. Math. France, 64 (1996), iv + 76 pp. · Zbl 0881.58060
[13] M. Kashiwara and P. Schapira, Ind-sheaves, Astérisque, 271 (2001), 136 pp. · Zbl 0993.32009
[14] Kashiwara, M.; Schapira, P., Microlocal study of ind-sheaves. I. micro-support and regularity, Astérisque, 284, 143-164, (2003) · Zbl 1053.35009
[15] M. Kashiwara and P. Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften, vol. 332, Springer, Berlin, 2006, x + 497 pp. · Zbl 1118.18001
[16] Kedlaya, K. S., Good formal structures for flat meromorphic connections, I: surfaces, Duke Math. J., 154, 343-418, (2010) · Zbl 1204.14010
[17] Kedlaya, K. S., Good formal structures for flat meromorphic connections, II: excellent schemes, J. Am. Math. Soc., 24, 183-229, (2011) · Zbl 1282.14037
[18] H. Majima, Asymptotic Analysis for Integrable Connections with Irregular Singular Points, Lecture Notes in Mathematics, vol. 1075, Springer, Berlin, 1984, xiv + 249 pp. · Zbl 0546.58003
[19] Mochizuki, T., Good formal structure for meromorphic flat connections on smooth projective surfaces, No. 54, 223-253, (2009), Tokyo · Zbl 1183.14027
[20] T. Mochizuki, Wild harmonic bundles and wild pure twistor \(\mathcal{D}\)-modules, Astérisque, 340, (2011), x + 607 pp. · Zbl 1245.32001
[21] Morando, G., An existence theorem for tempered solutions of \(\mathcal{D}\)-modules on complex curves, Publ. Res. Inst. Math. Sci., 43, 625-659, (2007) · Zbl 1155.32018
[22] Morando, G., Preconstructibility of tempered solutions of holonomic \(\mathcal{D}\)-modules, Int. Math. Res. Not., 4, 1125-1151, (2014) · Zbl 1396.32006
[23] C. Sabbah, Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque, 263, (2000), viii + 190 pp.
[24] C. Sabbah, Introduction to Stokes Structures, Lecture Notes in Mathematics, vol. 2060, Springer, Berlin, 2013, xiv + 249 pp. · Zbl 1260.34002
[25] D. Tamarkin, Microlocal condition for non-displaceability, 2008, 93 pp., arXiv:0809.1584v1. · Zbl 1416.35019
[26] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Pure and Applied Mathematics, vol. XIV, Wiley, New York, 1965, ix + 362 pp. · Zbl 0133.35301
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