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Three Lectures on Algebraic Microlocal Analysis

  • Pierre Schapira
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 269)

Abstract

This is a survey talk with some historical comments. I will first explain the notions of Sato’s hyperfunctions and microfunctions, at the origin of the story, and I will describe the Sato’s microlocalization functor which was first motivated by problems of analysis. Then I will briefly recall the main features of the microlocal theory of sheaves with emphasize on the functor \(\mu \)hom which will be an essential tool in the sequel. Then, I will construct the microlocal Euler class associated with trace kernels. This construction applies in particular to constructible sheaves on real manifolds and \(\mathscr {D}\)-modules (or more generally, elliptic pairs) on complex manifolds. Finally, I will first recall the construction of the sheaves of holomorphic functions with temperate growth or with exponential decay. These are not sheaves on the usual topology, but ind-sheaves, or else, sheaves on the subanalytic site. I will explain how these objects appear naturally in the study of irregular holonomic \(\mathscr {D}\)-modules.

Keywords

Microlocal sheaf theory \(\mathscr {D}\)-modules Hyperfunctions Index theorem Hochschild homologyMSC14F05 35A27 53D37 

References

  1. 1.
    Bloch, S., Esnault, H.: Homology for irregular connections. J. Théor. Nombres Bordeaux 16, 357–371 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bressler, P., Gorokhovsky, A., Nest, R., Tsygan, B.: Deformation quantization of gerbes. Adv. Math. 214, 230–266 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bressler, P., Nest, R., Tsygan, B.: Riemann-Roch theorems via deformation quantization. I, II. Adv. Math. 167, 1–25, 26–73 (2002)Google Scholar
  4. 4.
    Bros, J., Iagolnitzer, D.: Causality and local analyticity: mathematical study. Ann. Inst. Fourier 18, 147–184 (1973)Google Scholar
  5. 5.
    Brylinski, J-L., Getzler, E.: The homology of algebras of pseudodifferential symbols and the noncommutative residue, \(K\)-Theory 1, 385–403 (1987)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Caldararu, A.: The Mukai pairing II: the Hochschild-Kostant-Rosenberg isomorphism. Adv. Math. 194, 34–66 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Caldararu, A., Willerton, S.: The Mukai pairing I: a categorical approach. New York J. Math. 16 (2010). arXiv:0707.2052
  8. 8.
    D’Agnolo, A., Kashiwara, M.: Riemann-Hilbert correspondence for irregular holonomic systems. Publ. Math. Inst. Hautes Etudes Sci. 123, 69–197 (2016). arXiv:1311.2374
  9. 9.
    Fang, B., Liu, M., Treumann, D., Zaslow, E.: The coherent-constructible correspondence and Fourier-Mukai transforms. Acta Math. Sin. (Engl. Ser.) 27(2), 275–308 (2011). arXiv:1009.3506MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gabber, O.: The integrability of the characteristic variety. Am. J. Math. 103, 445–468 (1981)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Grivaux, J.: On a conjecture of Kashiwara relating Chern and Euler classes of \({\mathscr {O}}\)-modules. J. Differ. Geom. 267–275 (2012). arXiv:0910.5384
  12. 12.
    Guillermou, S., Schapira, P.: Construction of sheaves on the subanalytic site. Astrique 234, 1–60 (2016). arXiv:1212.4326
  13. 13.
    Hien, M.: Periods for flat algebraic connections. Invent. Math. 178, 1–22 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hörmander, L.: The analysis of linear partial differential operators. Grundlehren der Math. Wiss. 256 (1983). SpringerGoogle Scholar
  15. 15.
    Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs, Oxford (2006)Google Scholar
  16. 16.
    Kashiwara, M.: On the holonomic systems of linear differential equations, II. Invent. Math. 49, 121–135 (1978)Google Scholar
  17. 17.
    Kashiwara, M.: Faisceaux constructibles et systèmes holonômes d’équations aux dérivées partielles linéaires à points singuliers réguliers. Séminaire Goulaouic-Schwartz, 1979–1980 (French), Exp. No. 19 École Polytech., Palaiseau, (1980)Google Scholar
  18. 18.
    Kashiwara, M.: The Riemann-Hilbert problem for holonomic systems. Publ. Res. Inst. Math. Sci. 20, 319–365 (1984)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kashiwara, M.: Index theorem for constructible sheaves, Differential systems and singularities. Astrisque 130, Soc. Math. France, 193–209 (1985)Google Scholar
  20. 20.
    Kashiwara, M.: Letter to P. Schapira, unpublished, 18/11/1991Google Scholar
  21. 21.
    Kashiwara, M.: D-modules and Microlocal Calculus. Translations of mathematical monographs, vol. 217. American Math. Soc. (2003)Google Scholar
  22. 22.
    Kashiwara, M., Schapira, P.: Micro-support des faisceaux: applications aux modules différentiels. C. R. Acad. Sci. Paris série I Math 295, 487–490 (1982)Google Scholar
  23. 23.
    Kashiwara, M., Schapira, P.: Microlocal Study of Sheaves, Astérisque, p. 128. Soc. Math. France (1985)Google Scholar
  24. 24.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Grundlehren der Math. Wiss., vol. 292. Springer, Berlin (1990)Google Scholar
  25. 25.
    Kashiwara, M., Schapira, P.: Moderate and Formal Cohomology Associated With Constructible Sheaves, vol. 64, p. iv+76. Mém. Soc. Math. France (1996)CrossRefGoogle Scholar
  26. 26.
    Kashiwara, M., Schapira, P.: Integral transforms with exponential kernels and Laplace transform. J. AMS 10, 939–972 (1997)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kashiwara, M., Schapira, P.: Ind-sheaves, vol. 271, p. 136. Astérisque (2001)Google Scholar
  28. 28.
    Kashiwara, M., Schapira, P.: Microlocal study of ind-sheaves. I. Micro-support and regularity. Astérisque 284, 143–164 (2003)Google Scholar
  29. 29.
    Kashiwara, M., Schapira, P.: Categories and Sheaves, vol. 332. Grundlehren der Math. Wiss. (2006)Google Scholar
  30. 30.
    Kashiwara, M., Schapira, P.: Deformation Quantization Modules, vol. 345. Astérisque Soc. Math. France (2012). arXiv:1003.3304
  31. 31.
    Kashiwara, M., Schapira, P.: Microlocal Euler classes and Hochschild homology. J. Inst. Math. Jussieu 13, 487–516 (2014). arXiv:1203.4869MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kashiwara, M., Schapira, P.: Irregular holonomic kernels and Laplace transform. Selecta Math. 22, 55–101 (2016). arXiv:1402.3642MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kashiwara, M., Schapira, P.: Regular and irregular holonomic D-modules. Lecture note series, vol. 433. London Math Society (2016)Google Scholar
  34. 34.
    Kedlaya, K.S.: Good formal structures for flat meromorphic connections, I: surfaces. Duke Math. J. 154, 343–418 (2010)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kedlaya, K.S.: Good formal structures for flat meromorphic connections, II: Excellent schemes. J. Am. Math. Soc. 24, 183–229 (2011)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Laumon, G.: Sur la catégorie dérivée des D-modules filtérs, LNM 1016, Springer, 151—237 (1983)Google Scholar
  37. 37.
    Lojaciewicz, S.: Sur le problème de la division. Studia Math. 8, 87–156 (1961)Google Scholar
  38. 38.
    Malgrange, B.: Ideals of Differentiable Functions. Tata Institute, Oxford University Press (1967)Google Scholar
  39. 39.
    Martineau, A.: Théorèmes sur le prolongement analytique du type Edge of the Wedge. Sem. Bourbaki, 340 (1967/68)Google Scholar
  40. 40.
    Mochizuki, T.: Good formal structure for meromorphic flat connections on smooth projective surfaces. In: Algebraic Analysis and Around, Advances Studies in Pure Math, vol. 54, pp. 223–253. Math. Soc. Japan (2009)Google Scholar
  41. 41.
    Morando, G.: Temperate holomorphic solutions of \({\mathscr {D}}\)-modules on curves and formal invariants. Ann. Inst. Fourier (Grenoble) 59, 1611–1639 (2009)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Morando, G.: Constructibility of tempered solutions of holonomic D-modules. arXiv:1311.6621
  43. 43.
    Nadler, D.: Microlocal branes are constructible sheaves. Selecta Math. 15, 563–619 (2009)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22, 233–286 (2009)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Prelli, L.: Sheaves on subanalytic sites. Rendiconti del Seminario Matematico dell’Universit di Padova 120, 167–216 (2008)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Prelli, L.: Microlocalization of Subanalytic Sheaves. Mém. Soc. Math, France (2012)Google Scholar
  47. 47.
    Ramadoss, A.C.: The relative Riemann-Roch theorem from Hochschild homology. New York J. Math. 14, 643–717 (2008). arXiv:math/0603127
  48. 48.
    Ramadoss, A.C.: The Mukai pairing and integral transforms in Hochschild homology. Moscow Math. J. 10, 629–645 (2010)Google Scholar
  49. 49.
    Sabbah, C.: Théorie de Hodge et correspondance de Hitchin-Kobayashi sauvage, d’après T. Mochizuki, Sém. Bourbaki 1050 (2011–2012)Google Scholar
  50. 50.
    Sato, M.: Theory of hyperfunctions, I and II. J. Fac. Sci. Univ. Tokyo 8, 139–193; 487–436 (1959–1960)Google Scholar
  51. 51.
    S-G-A 4, Sém. Géom. Alg. (1963–64) Artin, M., Grothendieck, A., Verdier, J-L.: Théorie des topos et cohomologie étale des schémas, Lecture Notes in Math. 269, 270, 305 (1972/73)Google Scholar
  52. 52.
    Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and pseudo-differential equations. In: Komatsu, (ed.) Proceedings of the Hyperfunctions and Pseudo-differential Equations. Lecture notes in mathematics, vol. 287, pp. 265–529. Springer, Berlin (1973). Katata 1971Google Scholar
  53. 53.
    Schapira, P.: Mikio Sato, a visionary of mathematics. Not. AMS 2, 54 (2007)Google Scholar
  54. 54.
    Schapira, P.: Triangulated categories for the analysts. In: Triangulated Categories. London Math. Soc. LNS, vol. 375, pp. 371–389. Cambridge University Press, Cambridge (2010)Google Scholar
  55. 55.
    Schapira, P., Schneiders, J-P.: Index theorem for elliptic pairs. Astrisque Soc. Math. France 224 (1994)Google Scholar
  56. 56.
    Schapira, P., Schneiders, J.-P.: Derived category of filtered objects. Astrique 234, 1–60 (2016). arXiv:math.AG:1306.1359
  57. 57.
    Schneiders, J.-P.: Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. (N.S.) 76 (1999)Google Scholar
  58. 58.
    Sjöstrand, J.: Singularités analytiques microlocales, Astérisque. Soc. Math. France 95 (1982)Google Scholar
  59. 59.
    Van den Bergh, M.: A relation between Hochschild homology and cohomology for Gorenstein rings. Proc. Am. Math. Soc. 126, 1345–1348 (1998); Erratum 130, 2809–2810 (2002)Google Scholar

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Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuSorbonne Universités, UPMC Univ Paris 6ParisFrance

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