Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the codimension-three conjecture

Masaki Kashiwara and Kari Vilonen

Full-text: Open access

Abstract

The codimension-three conjecture states that any regular holonomic module extends uniquely beyond an analytic subset with codimension equal to or larger than three. We give a sketch of a proof of this conjecture.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 86, Number 9 (2010), 154-158.

Dates
First available in Project Euclid: 8 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.pja/1289226853

Digital Object Identifier
doi:10.3792/pjaa.86.154

Mathematical Reviews number (MathSciNet)
MR2780007

Zentralblatt MATH identifier
1225.32018

Subjects
Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 32W25: Pseudodifferential operators in several complex variables

Keywords
Codimension-three conjecture regular holonomic modules microlocal perverse sheaf

Citation

Kashiwara, Masaki; Vilonen, Kari. On the codimension-three conjecture. Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 9, 154--158. doi:10.3792/pjaa.86.154. https://projecteuclid.org/euclid.pja/1289226853.


Export citation

References

  • A. A. Beĭ linson, How to glue perverse sheaves, in $K$-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math., 1289, Springer, Berlin, (1987), 42–51.
  • Jan-Eric Björk, Analytic $\mathcal{D}$-modules and applications, Mathematics and its Applications, 247. Kluwer Academic Publishers Group, 1993, 581 pp.
  • L. Bungart, On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers, Topology 7 (1967), 55–68.
  • Adrien Douady, Prolongement de faisceaux analytiques coherents (Travaux de Trautmann, Frisch-Guenot et Siu), Séminaire Bourbaki, 1969/70.
  • S. Gelfand, R. MacPherson and K. Vilonen, Perverse sheaves and quivers, Duke Math. J. 83 (1996), no. 3, 621–643.
  • S. Gelfand, R. MacPherson and K. Vilonen, Micro-local perverse sheaves..
  • M. Kashiwara, Introduction to microlocal analysis, Enseign. Math. (2) 32 (1986), no. 3–4, 227–259.
  • M. Kashiwara, $D$-modules and microlocal calculus, Translated from the 2000 Japanese original by Mutsumi Saito, Amer. Math. Soc., Providence, RI, 2003.
  • M. Kashiwara and T. Kawai, On holonomic systems of microdifferential equations. III. Systems with regular singularities, Publ. Res. Inst. Math. Sci. 17 (1981), no. 3, 813–979.
  • M. Kashiwara and P. Schapira, Micro-support des faisceaux: application aux modules différentiels, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 8, 487–490.
  • M. Kashiwara and P. Schapira, Sheaves on manifolds, Springer, Berlin, 1990.
  • M. Kashiwara and P. Schapira, Deformation quantization modules, arXiv:1003.3304.
  • R. MacPherson and K. Vilonen, Elementary construction of perverse sheaves, Invent. Math. 84 (1986), no. 2, 403–435.
  • D. Popescu, On a question of Quillen, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 45(93) (2002), no. 3–4, (2003), 209–212.
  • D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171.
  • M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, in Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., 287, Springer, Berlin, (1973), 265–529.
  • P. Schapira, Microdifferential systems in the complex domain, Grundlehren der Mathematischen Wissenschaften 269 Springer, Berlin, (1985).
  • R. G. Swan, Néron-Popescu desingularization, in Algebra and geometry (Taipei, 1995), 135–192, Int. Press, Cambridge, MA.
  • I. Waschkies, The stack of microlocal perverse sheaves, Bull. Soc. Math. France 132 (2004), no. 3, 397–462.