Microdifferential systems and the codimension-three conjecture

Abstract

In this paper we give a proof of a fundamental conjecture, the codimension-three conjecture, for microdifferential holonomic systems. The conjecture states that any (regular) holonomic module extends uniquely beyond an analytic subset that is at least of codimension three in its support. Our result can also be interpreted from a topological point of view as a statement about microlocal perverse sheaves. However, our proof is entirely in the context of microdifferential holonomic systems.

  • [K0] Go to document M. Kashiwara, "The Riemann-Hilbert problem for holonomic systems," Publ. Res. Inst. Math. Sci., vol. 20, iss. 2, pp. 319-365, 1984.
    @article{K0,
      author = {Kashiwara, Masaki},
      journal = {Publ. Res. Inst. Math. Sci.},
      number = {2},
      pages = {319--365},
      title = {The {R}iemann-{H}ilbert problem for holonomic systems},
      volume = {20},
      year = {1984},
      doi = {10.2977/prims/1195181610},
      issn = {0034-5318},
      }
  • [SKK] M. Sato, T. Kawai, and M. Kashiwara, "Microfunctions and pseudo-differential equations," in Hyperfunctions and Pseudo-Differential Equations, New York: Springer-Verlag, 1973, vol. 287, pp. 265-529.
    @incollection{SKK, address = {New York},
      author = {Sato, Mikio and Kawai, Takahiro and Kashiwara, Masaki},
      booktitle = {Hyperfunctions and Pseudo-Differential Equations},
      pages = {265--529},
      publisher = {Springer-Verlag},
      series = {Lecture Notes in Math.},
      title = {Microfunctions and pseudo-differential equations},
      volume = {287},
      year = {1973},
      }
  • [KS1] M. Kashiwara and P. Schapira, "Micro-support des faisceaux: application aux modules différentiels," C. R. Acad. Sci. Paris Sér. I Math., vol. 295, iss. 8, pp. 487-490, 1982.
    @article{KS1,
      author = {Kashiwara, Masaki and Schapira, Pierre},
      journal = {C. R. Acad. Sci. Paris Sér. I Math.},
      number = {8},
      pages = {487--490},
      title = {Micro-support des faisceaux: application aux modules différentiels},
      volume = {295},
      year = {1982},
      issn = {0249-6321},
      }
  • [KS2] M. Kashiwara and P. Schapira, Sheaves on Manifolds, New York: Springer-Verlag, 1990, vol. 292.
    @book{KS2, address = {New York},
      author = {Kashiwara, Masaki and Schapira, Pierre},
      pages = {x+512},
      publisher = {Springer-Verlag},
      series = {Grundl. Math. Wissen.},
      title = {Sheaves on Manifolds},
      volume = {292},
      year = {1990},
      isbn = {3-540-51861-4},
      }
  • [A1] E. Andronikof, "A microlocal version of the Riemann-Hilbert correspondence," Sūrikaisekikenkyūsho Kōkyūroku, iss. 845, pp. 1-7, 1993.
    @article{A1,
      author = {Andronikof, Emmanuel},
      journal = {Sūrikaisekikenkyūsho Kōkyūroku},
      note = {Microlocal Geometry (Kyoto, 1992)},
      number = {845},
      pages = {1--7},
      title = {A microlocal version of the {R}iemann-{H}ilbert correspondence},
      year = {1993},
      }
  • [A2] Go to document E. Andronikov, "Microlocalization of perverse sheaves," J. Math. Sci., vol. 82, iss. 6, pp. 3754-3758, 1996.
    @article{A2,
      author = {Andronikov, E.},
      journal = {J. Math. Sci.},
      number = {6},
      pages = {3754--3758},
      title = {Microlocalization of perverse sheaves},
      volume = {82},
      year = {1996},
      doi = {10.1007/BF02362631},
      issn = {1072-3374},
      }
  • [W] I. Waschkies, "The stack of microlocal perverse sheaves," Bull. Soc. Math. France, vol. 132, iss. 3, pp. 397-462, 2004.
    @article{W,
      author = {Waschkies, Ingo},
      journal = {Bull. Soc. Math. France},
      number = {3},
      pages = {397--462},
      title = {The stack of microlocal perverse sheaves},
      volume = {132},
      year = {2004},
      issn = {0037-9484},
      }
  • [Be] Go to document A. A. Beuilinson, "How to glue perverse sheaves," in $K$-Theory, Arithmetic and Geometry, New York: Springer-Verlag, 1987, vol. 1289, pp. 42-51.
    @incollection{Be, address = {New York},
      author = {Be{\u\i}linson, A. A.},
      booktitle = {{$K$}-Theory, Arithmetic and Geometry},
      pages = {42--51},
      publisher = {Springer-Verlag},
      series = {Lecture Notes in Math.},
      title = {How to glue perverse sheaves},
      volume = {1289},
      year = {1987},
      doi = {10.1007/BFb0078366},
      }
  • [K1] M. Kashiwara, "Introduction to microlocal analysis," Enseign. Math., vol. 32, iss. 3-4, pp. 227-259, 1986.
    @article{K1,
      author = {Kashiwara, Masaki},
      journal = {Enseign. Math.},
      number = {3-4},
      pages = {227--259},
      title = {Introduction to microlocal analysis},
      volume = {32},
      year = {1986},
      issn = {0013-8584},
      }
  • [KK] Go to document M. Kashiwara and T. Kawai, "On holonomic systems of microdifferential equations. III. Systems with regular singularities," Publ. Res. Inst. Math. Sci., vol. 17, iss. 3, pp. 813-979, 1981.
    @article{KK,
      author = {Kashiwara, Masaki and Kawai, Takahiro},
      journal = {Publ. Res. Inst. Math. Sci.},
      number = {3},
      pages = {813--979},
      title = {On holonomic systems of microdifferential equations. {III}. {S}ystems with regular singularities},
      volume = {17},
      year = {1981},
      doi = {10.2977/prims/1195184396},
      issn = {0034-5318},
      }
  • [MV] Go to document R. MacPherson and K. Vilonen, "Elementary construction of perverse sheaves," Invent. Math., vol. 84, iss. 2, pp. 403-435, 1986.
    @article{MV,
      author = {MacPherson, Robert and Vilonen, Kari},
      journal = {Invent. Math.},
      number = {2},
      pages = {403--435},
      title = {Elementary construction of perverse sheaves},
      volume = {84},
      year = {1986},
      doi = {10.1007/BF01388812},
      issn = {0020-9910},
      }
  • [GMV1] Go to document S. Gelfand, R. MacPherson, and K. Vilonen, "Perverse sheaves and quivers," Duke Math. J., vol. 83, iss. 3, pp. 621-643, 1996.
    @article{GMV1,
      author = {Gelfand, Sergei and MacPherson, Robert and Vilonen, Kari},
      journal = {Duke Math. J.},
      number = {3},
      pages = {621--643},
      title = {Perverse sheaves and quivers},
      volume = {83},
      year = {1996},
      doi = {10.1215/S0012-7094-96-08319-2},
      issn = {0012-7094},
      }
  • [GMV2] S. Gelfand, R. MacPherson, and K. Vilonen, Micro-local perverse sheaves.
    @misc{GMV2,
      author = {Gelfand, Sergei and MacPherson, Robert and Vilonen, Kari},
      title = {Micro-local perverse sheaves},
      }
  • [M] B. Malgrange, "On irregular holonomic $\mathcal{D}$-modules," in Éléments de la Théorie des Systèmes Différentiels Géométriques, Soc. Math. France, Paris, 2004, vol. 8, pp. 391-410.
    @incollection{M,
      author = {Malgrange, Bernard},
      booktitle = {\'{E}léments de la Théorie des Systèmes Différentiels Géométriques},
      pages = {391--410},
      publisher = {Soc. Math. France, Paris},
      series = {Sémin. Congr.},
      title = {On irregular holonomic {$\mathcal{D}$}-modules},
      volume = {8},
      year = {2004},
      }
  • [T] Go to document G. Trautmann, "Ein Kontinuitätssatz für die Fortsetzung kohärenter analytischer Garben," Arch. Math. $($Basel$)$, vol. 18, pp. 188-196, 1967.
    @article{T,
      author = {Trautmann, G{ü}nther},
      journal = {Arch. Math. $($Basel$)$},
      pages = {188--196},
      title = {Ein {K}ontinuitätssatz für die {F}ortsetzung kohärenter analytischer {G}arben},
      volume = {18},
      year = {1967},
      doi = {10.1007/BF01899645},
      issn = {0003-889X},
      }
  • [FG] Go to document J. Frisch and J. Guenot, "Prolongement de faisceaux analytiques cohérents," Invent. Math., vol. 7, pp. 321-343, 1969.
    @article{FG,
      author = {Frisch, J. and Guenot, J.},
      journal = {Invent. Math.},
      pages = {321--343},
      title = {Prolongement de faisceaux analytiques cohérents},
      volume = {7},
      year = {1969},
      doi = {10.1007/BF01425539},
      issn = {0020-9910},
      }
  • [Siu1] Go to document Y. Siu, "Extending coherent analytic sheaves," Ann. of Math., vol. 90, pp. 108-143, 1969.
    @article{Siu1,
      author = {Siu, Yum-tong},
      journal = {Ann. of Math.},
      pages = {108--143},
      title = {Extending coherent analytic sheaves},
      volume = {90},
      year = {1969},
      doi = {10.2307/1970684},
      issn = {0003-486X},
      }
  • [Bu] Go to document L. Bungart, "On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers," Topology, vol. 7, pp. 55-68, 1967.
    @article{Bu,
      author = {Bungart, Lutz},
      journal = {Topology},
      pages = {55--68},
      title = {On analytic fiber bundles. {I}. {H}olomorphic fiber bundles with infinite dimensional fibers},
      volume = {7},
      year = {1967},
      doi = {10.1016/0040-9383(86)90015-7},
      issn = {0040-9383},
      }
  • [ST] Y. Siu and G. Trautmann, Gap-Sheaves and Extension of Coherent Analytic Subsheaves, New York: Springer-Verlag, 1971, vol. 172.
    @book{ST, address = {New York},
      author = {Siu, Yum-tong and Trautmann, G{ü}nther},
      pages = {v+172},
      publisher = {Springer-Verlag},
      series = {Lecture Notes in Math.},
      title = {Gap-Sheaves and Extension of Coherent Analytic Subsheaves},
      volume = {172},
      year = {1971},
      }
  • [P] D. Popescu, "On a question of Quillen," Bull. Math. Soc. Sci. Math. Roumanie, vol. 45(93), iss. 3-4, pp. 209-212 (2003), 2002.
    @article{P,
      author = {Popescu, Dorin},
      journal = {Bull. Math. Soc. Sci. Math. Roumanie},
      number = {3-4},
      pages = {209--212 (2003)},
      title = {On a question of {Q}uillen},
      volume = {45(93)},
      year = {2002},
      issn = {1220-3874},
      }
  • [BR] Go to document S. M. Bhatwadekar and R. A. Rao, "On a question of Quillen," Trans. Amer. Math. Soc., vol. 279, iss. 2, pp. 801-810, 1983.
    @article{BR,
      author = {Bhatwadekar, S. M. and Rao, R. A.},
      journal = {Trans. Amer. Math. Soc.},
      number = {2},
      pages = {801--810},
      title = {On a question of {Q}uillen},
      volume = {279},
      year = {1983},
      doi = {10.2307/1999568},
      issn = {0002-9947},
      }
  • [KV] Go to document M. Kashiwara and K. Vilonen, "On the codimension-three conjecture," Proc. Japan Acad. Ser. A Math. Sci., vol. 86, iss. 9, pp. 154-158, 2010.
    @article{KV,
      author = {Kashiwara, Masaki and Vilonen, Kari},
      journal = {Proc. Japan Acad. Ser. A Math. Sci.},
      number = {9},
      pages = {154--158},
      title = {On the codimension-three conjecture},
      volume = {86},
      year = {2010},
      doi = {10.3792/pjaa.86.154},
      issn = {0386-2194},
      }
  • [Sch] Go to document P. Schapira, Microdifferential Systems in the Complex Domain, New York: Springer-Verlag, 1985, vol. 269.
    @book{Sch, address = {New York},
      author = {Schapira, Pierre},
      pages = {x+214},
      publisher = {Springer-Verlag},
      series = {Grundl. Math. Wissen.},
      title = {Microdifferential Systems in the Complex Domain},
      volume = {269},
      year = {1985},
      doi = {10.1007/978-3-642-61665-5},
      isbn = {3-540-13672-X},
      }
  • [K2] M. Kashiwara, $D$-Modules and Microlocal Calculus, Amer. Math. Soc., 2003, vol. 217.
    @book{K2,
      author = {Kashiwara, Masaki},
      note = {Translated from the 2000 Japanese original by Mutsumi Saito, \emph{Iwanami Series in Modern Math.}},
      pages = {xvi+254},
      publisher = {Amer. Math. Soc.},
      series = {Trans. Math. Monogr.},
      title = {{$D$}-Modules and Microlocal Calculus},
      volume = {217},
      year = {2003},
      isbn = {0-8218-2766-9},
      }
  • [Bj] J. Björk, Analytic $\mathcal{D}$-modules and Applications, Dordrecht: Kluwer Academic Publishers Group, 1993, vol. 247.
    @book{Bj, address = {Dordrecht},
      author = {Bj{ö}rk, Jan-Eric},
      publisher = {Kluwer Academic Publishers Group},
      series = {Math. Appl.},
      title = {Analytic {$\mathcal{D}$}-modules and Applications},
      volume = {247},
      year = {1993},
      }
  • [S] R. G. Swan, "Néron-Popescu desingularization," in Algebra and Geometry, Cambridge, MA: Internat. Press, 1998, pp. 135-192.
    @incollection{S, address = {Cambridge, MA},
      author = {Swan, Richard G.},
      booktitle = {Algebra and Geometry},
      number = {2},
      pages = {135--192},
      publisher = {Internat. Press},
      series = {Lect. Algebra Geom.},
      title = {Néron-{P}opescu desingularization},
      year = {1998},
      }
  • [Q] Go to document D. Quillen, "Projective modules over polynomial rings," Invent. Math., vol. 36, pp. 167-171, 1976.
    @article{Q,
      author = {Quillen, Daniel},
      journal = {Invent. Math.},
      pages = {167--171},
      title = {Projective modules over polynomial rings},
      volume = {36},
      year = {1976},
      doi = {10.1007/BF01390008},
      issn = {0020-9910},
      }
  • [KS3] Go to document M. Kashiwara and P. Schapira, Deformation Quantization Modules, , 2012, vol. 345.
    @book{KS3,
      author = {Kashiwara, Masaki and Schapira, Pierre},
      pages = {xii+147},
      series = {Astérisque},
      title = {Deformation Quantization Modules},
      volume = {345},
      year = {2012},
      doi = {10.1007/978-3-662-02661-8},
      isbn = {978-2-85629-345-4},
      issn = {0303-1179},
      }

Authors

Masaki Kashiwara

Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan and
Seoul National University, Seoul, Korea

Kari Vilonen

Northwestern University, Evanston, IL and
Helsinki University, Helsinki, Finland