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Riemann–Hilbert correspondence for irregular holonomic \({\mathscr{D}}\)-modules

Abstract

This is a survey paper on the Riemann–Hilbert correspondence on (irregular) holonomic \({\mathscr{D}}\)-modules, based on the 16th Takagi Lectures (2015/11/28). In this paper, we use subanalytic sheaves, an analogous notion to the one of indsheaves.

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References

  1. SGA4

    M. Artin, A. Grothendieck and J.L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA4), Lecture Notes in Math., vol. 1, 269 (1972); vol. 2, 270 (1972); vol. 3, 305 (1973); Springer-Verlag.

  2. BM88

    Bierstone E., Milman P.D.: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. 67, 5–42 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  3. Bj93

    J.-E. Björk, Analytic \({{\mathscr{D}}}\)-modules and Applications, Math. Appl., 247, Kluwer Academic Publishers, Dordrecht, 1993.

  4. DK13

    A. D’Agnolo and M. Kashiwara, Riemann–Hilbert correspondence for holonomic \({{\mathscr{D}}}\)-modules, to appear in Publ. Math. Inst. Hautes Études Sci.; preprint, arXiv:1311.2374.

  5. DK15

    A. D’Agnolo and M. Kashiwara, Enhanced perversities, preprint, arXiv:1509.03791.

  6. De70

    P. Deligne, Équations Différentielles à Points Singuliers réguliers, Lecture Notes in Math., 163, Springer-Verlag, 1970.

  7. DMR07

    P. Deligne, B. Malgrange and J.-P. Ramis, Singularités Irrégulières, Correspondance Et Documents, Doc. Math. (Paris), 5, Soc. Math. France, 2007.

  8. Gabb81

    Gabber O.: The integrability of the characteristic variety. Amer. J. Math. 103, 445–468 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  9. Gabr68

    A.M. Gabrièlov, Projections of semianalytic sets. (Russian), Funkcional. Anal. i Priložen., 2, no. 4 (1968), 18–30.

  10. Hi73

    H. Hironaka, Subanalytic sets, In: Number Theory, Algebraic Geometry and Commutative Algebra; in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453–493.

  11. HTT08

    R. Hotta, K. Takeuchi and T. Tanisaki, \({D}\)-modules, Perverse Sheaves, and Representation Theory, Progr. Math., 236, Birkhäuser, Boston, MA, 2008.

  12. Ka70

    M. Kashiwara, Algebraic study of systems of partial differential equations, Master’s thesis, Univ. of Tokyo, 1970; Mém. Soc. Math. France (N.S.), 63, Soc. Math. France, 1995.

  13. Ka75

    Kashiwara M.: On the maximally overdetermined system of linear differential equations. I. Publ. Res. Inst. Math. Sci. 10, 563–579 (1974)

    MathSciNet  Article  MATH  Google Scholar 

  14. Ka78

    Kashiwara M.: On the holonomic systems of linear differential equations.II. Invent. Math. 49, 121–135 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  15. Ka80

    M. Kashiwara, Faisceaux constructibles et systèmes holonômes d’équations aux dérivées partielles linéaires à points singuliers réguliers, In: Séminaire Goulaouic–Schwartz, 1979–1980, 19, École Polytech., Palaiseau, 1980.

  16. Ka84

    Kashiwara M.: The Riemann–Hilbert problem for holonomic systems. Publ. Res. Inst. Math. Sci. 20, 319–365 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  17. Ka03

    M. Kashiwara, \({D}\)-modules and Microlocal Calculus, Transl. Math. Monogr., 217, Amer. Math. Soc., Providence, RI, 2003.

  18. KK81

    Kashiwara M., Kawai T.: On holonomic systems of microdifferential equations. III. Systems with regular singularities. Publ. Res. Inst. Math. Sci. 17, 813–979 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  19. KS90

    M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss., 292, Springer-Verlag, 1990.

  20. KS96

    M. Kashiwara and P. Schapira, Moderate and Formal Cohomology Associated with Constructible Sheaves, Mém. Soc. Math. France (N.S.), 64, Soc. Math. France, 1996.

  21. KS01

    M. Kashiwara and P. Schapira, Ind-Sheaves, Astérisque, 271, Soc. Math. France, 2001.

  22. KS06

    M. Kashiwara and P. Schapira, Categories and Sheaves, Grundlehren Math. Wiss., 332, Springer-Verlag, 2006.

  23. KS14

    Kashiwara M., Schapira P.: Irregular holonomic kernels and Laplace transform. Selecta Math. (N.S.) 22, 55–109 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  24. KS15

    M. Kashiwara and P. Schapira, Lectures on Regular and Irregular Holonomic \({D}\)-modules, 2015, http://preprints.ihes.fr/2015/M/M-15-08.pdf; expanded version to appear in London Math. Soc. Lecture Note Ser.

  25. Ke10

    Kedlaya K.S.: Good formal structures for flat meromorphic connections. I: Surfaces. Duke Math. J. 154, 343–418 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  26. Ke11

    Kedlaya K.S.: Good formal structures for flat meromorphic connections. II: Excellent schemes. J. Amer. Math. Soc. 24, 183–229 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  27. Lo59

    Łojasiewicz S.: Sur le problème de la division. Studia Math. 8, 87–136 (1959)

    MATH  Google Scholar 

  28. Ma66

    B. Malgrange, Ideals of Differentiable Functions, Tata Inst. Fund. Res. Stud. Math., 3, Tata Inst. Fund. Res., Oxford Univ. Press, London, 1967.

  29. Mo09

    T. Mochizuki, Good formal structure for meromorphic flat connections on smooth projective surfaces, In: Algebraic Analysis and Around, Adv. Stud. Pure Math., 54, Math. Soc. Japan, Tokyo, 2009, pp. 223–253.

  30. Mo11

    T. Mochizuki, Wild Harmonic Bundles and Wild Pure Twistor \({{\mathscr{D}}}\)-modules, Astérisque, 340, Soc. Math. France, 2011.

  31. Pr08

    L. Prelli, Sheaves on subanalytic sites, Rend. Semin. Mat. Univ. Padova, 120 (2008), 167–216.

  32. Sa00

    C. Sabbah, Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque, 263, Soc. Math. France, 2000.

  33. Sa13

    C. Sabbah, Introduction to Stokes Structures, Lecture Notes in Math., 2060, Springer-Verlag, 2013.

  34. SKK73

    M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, In: Hyperfunctions and Pseudo-Differential Equations, Proc. of a Conference, Katata, 1971, (ed. H. Komatsu), Lecture Notes in Math., 287, Springer-Verlag, 1973, pp. 265–529.

  35. Sc86

    J.-P. Schneiders, Un théorème de dualité relative pour les modules différentiels, C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), 235–238.

  36. Ta08

    D. Tamarkin, Microlocal condition for non-displaceability, preprint, arXiv:0809.1584.

  37. VD98

    L. van den Dries, Tame Topology and O-minimal Structures, London Math. Soc. Lecture Note Ser., 248, Cambridge Univ. Press, 1998.

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Corresponding author

Correspondence to Masaki Kashiwara.

Additional information

This article is based on the 16th Takagi Lectures that the author delivered at the University of Tokyo on November 28 and 29, 2015.

The research was supported in part by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science.

Communicated by: Toshiyuki Kobayashi

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Kashiwara, M. Riemann–Hilbert correspondence for irregular holonomic \({\mathscr{D}}\)-modules. Jpn. J. Math. 11, 113–149 (2016). https://doi.org/10.1007/s11537-016-1564-7

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Keywords and phrases

  • irregular Riemann–Hilbert problem
  • irregular holonomic \({{\mathscr{D}}}\)-modules
  • ind-sheaves
  • subanalytic sheaves
  • Stokes phenomenon

Mathematics Subject Classification (2010)

  • 32C38
  • 35A27
  • 32S60