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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.

MSC:
 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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References:
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