On index theorems for linear ordinary differential operators

Michèle Loday-Richaud; Geneviève Pourcin

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 5, page 1379-1424
  • ISSN: 0373-0956

Abstract

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We introduce and study the sheaf of Deligne to describe singular points of a linear differential operator D and we develop a technique based on homological algebra to prove index theorems for D .As particular cases, we obtain index theorems for D acting in spaces of multisummable series and a new proof of the index theorem of Malgrange in the space of convergent power series and of the index theorems of Ramis in the spaces of Gevrey series.We compute the values of these indices in terms of the formal invariants and, namely, in terms of the “singular big points” of D .

How to cite

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Loday-Richaud, Michèle, and Pourcin, Geneviève. "On index theorems for linear ordinary differential operators." Annales de l'institut Fourier 47.5 (1997): 1379-1424. <http://eudml.org/doc/75268>.

@article{Loday1997,
abstract = {We introduce and study the sheaf of Deligne to describe singular points of a linear differential operator $D$ and we develop a technique based on homological algebra to prove index theorems for $D$.As particular cases, we obtain index theorems for $D$ acting in spaces of multisummable series and a new proof of the index theorem of Malgrange in the space of convergent power series and of the index theorems of Ramis in the spaces of Gevrey series.We compute the values of these indices in terms of the formal invariants and, namely, in terms of the “singular big points” of $D$.},
author = {Loday-Richaud, Michèle, Pourcin, Geneviève},
journal = {Annales de l'institut Fourier},
keywords = {linear ordinary differential operators; index; Gevrey series; multisummability; sheaf of Deligne},
language = {eng},
number = {5},
pages = {1379-1424},
publisher = {Association des Annales de l'Institut Fourier},
title = {On index theorems for linear ordinary differential operators},
url = {http://eudml.org/doc/75268},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Loday-Richaud, Michèle
AU - Pourcin, Geneviève
TI - On index theorems for linear ordinary differential operators
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 5
SP - 1379
EP - 1424
AB - We introduce and study the sheaf of Deligne to describe singular points of a linear differential operator $D$ and we develop a technique based on homological algebra to prove index theorems for $D$.As particular cases, we obtain index theorems for $D$ acting in spaces of multisummable series and a new proof of the index theorem of Malgrange in the space of convergent power series and of the index theorems of Ramis in the spaces of Gevrey series.We compute the values of these indices in terms of the formal invariants and, namely, in terms of the “singular big points” of $D$.
LA - eng
KW - linear ordinary differential operators; index; Gevrey series; multisummability; sheaf of Deligne
UR - http://eudml.org/doc/75268
ER -

References

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  1. [BV89] D.G. BABBITT, V.S. VARADARAJAN, Local moduli for meromorphic differential equations, Astérisque, 169-170 (1989). Zbl0683.34003MR91e:32017
  2. [BJL79] W. BALSER, W.B. JURKAT, D.A. LUTZ, A General Theory of Invariants for Meromorphic Differential Equations; Part I, Formal Invariants, Funkcialaj Ekvacioj, 22 (1979), 197-221. Zbl0434.34002MR83m:34003a
  3. [D77] P. DELIGNE, Lettre de P. Deligne à B. Malgrange (22 août 1977). 
  4. [D86] P. DELIGNE, Lettre de P. Deligne à J.-P. Ramis (7 janvier 1986). 
  5. [God] R. GODEMENT, Théorie des faisceaux, Hermann Paris 1958. Zbl0080.16201
  6. [Ive] B. IVERSEN, Cohomology of sheaves, Springer-Verlag 1986. Zbl0559.55001MR87m:14013
  7. [K71] H. KOMATSU, On the index of ordinary differential operators, J. Fac. Sc. Univ. Tokyo, I-A (1971), 379-398. Zbl0232.34026MR46 #2705
  8. [L-R94] M. LODAY-RICHAUD, Stokes phenomenon, multisummability and differential Galois groups, Ann. Inst. Fourier, Grenoble, 44-3 (1994), 849-906. Zbl0812.34004
  9. [M74] B. MALGRANGE, Sur les points singuliers des équations différentielles, L'Enseignement Mathématique, XX, 1-2 (1974), 147-176. Zbl0299.34011
  10. [M79] B. MALGRANGE, Remarques sur les équations différentielles à points singuliers irréguliers, in Lecture Notes in Mathematics, Équations différentielles et systèmes de Pfaff dans le plan complexe, édité par R. Gérard et J.-P. Ramis, 712, Springer-Verlag 1979, 77-86. Zbl0423.32014MR80k:14019
  11. [Mal] B. MALGRANGE, Équations différentielles à coefficients polynomiaux, Progress in Math., Birkhäuser, 1991. Zbl0764.32001MR92k:32020
  12. [M95] B. MALGRANGE, Sommation des séries divergentes, Expositiones Mathematicæ, 13, 2-3 (1995), 163-222. Zbl0836.40004MR96i:34125
  13. [MR92] B. MALGRANGE, J.-P. RAMIS, Fonctions multisommables, Ann. Inst. Fourier, Grenoble, 42, 1-2 (1992), 353-368. Zbl0759.34007MR93e:40007
  14. [R78] J.-P. RAMIS, Dévissage Gevrey, Astérisque, 59-60 (1978), 173-204. Zbl0409.34018MR81g:34010
  15. [R84] J.-P. RAMIS, Memoirs of the Am. Math. Soc., 48 (1984), 296. 
  16. [Was] W. WASOW, Asymptotic expansions for ordinary differential equations, Dover 1987, Intersc. Publ., 1965. Zbl0133.35301

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