The stack of microlocal perverse sheaves

Ingo Waschkies

Bulletin de la Société Mathématique de France (2004)

  • Volume: 132, Issue: 3, page 397-462
  • ISSN: 0037-9484

Abstract

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In this paper we construct the abelian stack of microlocal perverse sheaves on the projective cotangent bundle of a complex manifold. Following ideas of Andronikof we first consider microlocal perverse sheaves at a point using classical tools from microlocal sheaf theory. Then we will use Kashiwara-Schapira’s theory of analytic ind-sheaves to globalize our construction. This presentation allows us to formulate explicitly a global microlocal Riemann-Hilbert correspondence.

How to cite

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Waschkies, Ingo. "The stack of microlocal perverse sheaves." Bulletin de la Société Mathématique de France 132.3 (2004): 397-462. <http://eudml.org/doc/272464>.

@article{Waschkies2004,
abstract = {In this paper we construct the abelian stack of microlocal perverse sheaves on the projective cotangent bundle of a complex manifold. Following ideas of Andronikof we first consider microlocal perverse sheaves at a point using classical tools from microlocal sheaf theory. Then we will use Kashiwara-Schapira’s theory of analytic ind-sheaves to globalize our construction. This presentation allows us to formulate explicitly a global microlocal Riemann-Hilbert correspondence.},
author = {Waschkies, Ingo},
journal = {Bulletin de la Société Mathématique de France},
keywords = {sheaf; constructible sheaf; perverse sheaf; microlocal sheaf theory; ind-sheaf; $2$-limit; stack},
language = {eng},
number = {3},
pages = {397-462},
publisher = {Société mathématique de France},
title = {The stack of microlocal perverse sheaves},
url = {http://eudml.org/doc/272464},
volume = {132},
year = {2004},
}

TY - JOUR
AU - Waschkies, Ingo
TI - The stack of microlocal perverse sheaves
JO - Bulletin de la Société Mathématique de France
PY - 2004
PB - Société mathématique de France
VL - 132
IS - 3
SP - 397
EP - 462
AB - In this paper we construct the abelian stack of microlocal perverse sheaves on the projective cotangent bundle of a complex manifold. Following ideas of Andronikof we first consider microlocal perverse sheaves at a point using classical tools from microlocal sheaf theory. Then we will use Kashiwara-Schapira’s theory of analytic ind-sheaves to globalize our construction. This presentation allows us to formulate explicitly a global microlocal Riemann-Hilbert correspondence.
LA - eng
KW - sheaf; constructible sheaf; perverse sheaf; microlocal sheaf theory; ind-sheaf; $2$-limit; stack
UR - http://eudml.org/doc/272464
ER -

References

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  1. [1] E. Andronikof – « A microlocal version of the Riemann-Hilbert correspondence », Topol. Meth. Nonlin. Anal.4 (1994), p. 417–425. Zbl0839.32004MR1350980
  2. [2] —, Microlocalisation tempérée, Mém. Soc. Math. France (N.S.), vol. 57, Société Mathématique de France, Paris, 1994. 
  3. [3] A. Beilinson, J. Bernstein & P. Deligne – Faisceaux pervers, analyse et topologie sur les espaces singuliers, Astérisque, vol. 100, Soc. Math. France, 1982. Zbl0536.14011MR751966
  4. [4] A. D’Agnolo – « On the microlocal cut-off of sheaves », Topol. Meth. Nonlin. Anal.8 (1996), p. 161–167. Zbl0901.58070MR1485761
  5. [5] P. Gabriel – « Des catégories abéliennes », Bull. Soc. Math. France90 (1962), p. 323–448. Zbl0201.35602MR232821
  6. [6] S. Gelfand, R. MacPherson & K. Vilonen – « Microlocal Perverse Sheaves », anounced. 
  7. [7] M. Kashiwara – « The Riemann-Hilbert problem for holonomic systems », Publ. RIMS, Kyoto Univ. 20 (1984), p. 319–365. Zbl0566.32023MR743382
  8. [8] —, « Quantization of contact manifolds », Publ. RIMS, Kyoto Univ. 32 (1996), p. 1–7. Zbl0874.53027MR1384750
  9. [9] —, « Ind-Microlocalization », written by F.Ivorra and I.Waschkies following a manuscript of M.Kashiwara, to appear in Progress in Math., Birkhäuser. 
  10. [10] M. Kashiwara & T. Kawai – « On holonomic systems of microdifferential equations III », Publ. RIMS, Kyoto Univ. 17 (1981), p. 813–979. Zbl0505.58033MR650216
  11. [11] M. Kashiwara & P. Schapira – Sheaves on Manifolds, Springer, 1990. Zbl0709.18001MR1074006
  12. [12] —, Ind-sheaves, Astérisque, vol. 271, Société Mathématique de France, Paris, 2001. Zbl0993.32009
  13. [13] S. MacLane – Categories for the working mathematician, 2nd éd., Springer, 1998. Zbl0705.18001MR1712872
  14. [14] M. Sato, T. Kawai & M. Kashiwara – « Hyperfunctions and pseudo-differential equations », Proceedings Katata 1971 (H. Komatsu, éd.), Lecture Notes in Math., vol. 287, Springer, 1973, p. 265–529. Zbl0277.46039MR420735
  15. [15] R. Street – « Categorical structures », Handbook of algebra, vol. 1, North-Holland, 1996, p. 529–577. Zbl0854.18001MR1421811
  16. [16] R. Thomason – « Homotopy colimits in the category of small categories », Proc. Camb. Phil. Soc.85 (1979), p. 91–109. Zbl0392.18001MR510404

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