no. 4
Tempered solutions of 𝒟-modules on complex curves and formal invariants
Morando, Giovanni1
1 Università di Padova Dipartimento di Matematica Pura ed Applicata Via Trieste 63 35121 Padova (Italy) and Universidade de Lisboa Centro de Álgebra Av. Prof. Gama Pinto, 2 1649-003 Lisboa (Portugal)
Annales de l'Institut Fourier, Volume 59 (2009) no. 4, pp. 1611-1639.

Let X be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of 𝒟-modules on X induces a fully faithful functor on a subcategory of germs of formal holonomic 𝒟-modules. Further, given a germ ℳ of holonomic 𝒟-module, we obtain some results linking the subanalytic sheaf of tempered solutions of ℳ and the classical formal and analytic invariants of ℳ.

Soit X une courbe analytique complexe. Dans cet article nous démontrons que le faisceau sous-analytique des solutions holomorphes tempérées des 𝒟-modules sur X induit un foncteur pleinement fidèle sur une sous-catégorie des germes des 𝒟-modules holonomes formels. De plus, étant donné un germe ℳ de 𝒟-module holonome, nous obtenons des résultats qui lient le faisceau sous-analytique des solutions tempérées de ℳ avec les invariants formels et analytiques classiques de ℳ.

Received:
Accepted:
DOI: 10.5802/aif.2472
Classification: 34M35,  32B20,  34Mxx
Keywords: 𝒟-modules, irregular singularities, tempered holomorphic functions, subanalytic
Morando, Giovanni 1

1 Università di Padova Dipartimento di Matematica Pura ed Applicata Via Trieste 63 35121 Padova (Italy) and Universidade de Lisboa Centro de Álgebra Av. Prof. Gama Pinto, 2 1649-003 Lisboa (Portugal) 
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Morando, Giovanni. Tempered solutions of $\mathcal{D}$-modules on complex curves and formal invariants. Annales de l'Institut Fourier, Volume 59 (2009) no. 4, pp. 1611-1639. doi : 10.5802/aif.2472. https://aif.centre-mersenne.org/articles/10.5802/aif.2472/

[1] Babbitt, D. G.; Varadarajan, V. S. Local moduli for meromorphic differential equations, Astérisque (1989) no. 169-170, 217 pages | MR: 1014083 | Zbl: 0683.34003

[2] Bierstone, Edward; Milman, Pierre D. Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. (1988) no. 67, pp. 5-42 | Article | Numdam | MR: 972342 | Zbl: 0674.32002

[3] Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Volume 36, Springer-Verlag, Berlin, 1998 (Translated from the 1987 French original, Revised by the authors) | MR: 1659509 | Zbl: 0912.14023

[4] Chenciner, Alain Courbes algébriques planes, Publications Mathématiques de l’Université Paris VII [Mathematical Publications of the University of Paris VII], Volume 4, Université de Paris VII U.E.R. de Mathématiques, Paris, 1978 | Zbl: 0557.14016

[5] Coste, Michel An introduction to o-minimal geometry, Istituti Ed. e Poligrafici Intern., Università di Pisa, Dipartimento di Matematica, Pisa, 2000

[6] Deligne, Pierre; Malgrange, Bernard; Ramis, Jean-Pierre Singularités irrégulières, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 5, Société Mathématique de France, Paris, 2007 (Correspondance et documents. [Correspondence and documents]) | MR: 2387754 | Zbl: 1130.14001

[7] Kashiwara, M. Faisceaux constructibles et systèmes holonômes d’équations aux dérivées partielles linéaires à points singuliers réguliers, Séminaire Goulaouic-Schwartz, 1979–1980 (French), École Polytechnique, Palaiseau, 1980, 7 pages (Exp. No. 19) | Numdam | MR: 600704 | Zbl: 0444.58014

[8] Kashiwara, Masaki The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci., Volume 20 (1984) no. 2, pp. 319-365 | Article | MR: 743382 | Zbl: 0566.32023

[9] Kashiwara, Masaki D-modules and microlocal calculus, Translations of Mathematical Monographs, Volume 217, American Mathematical Society, Providence, RI, 2003 (Translated from the 2000 Japanese original by Mutsumi Saito, Iwanami Series in Modern Mathematics) | MR: 1943036 | Zbl: 1017.32012

[10] Kashiwara, Masaki; Schapira, Pierre Ind-sheaves, Astérisque (2001) no. 271, 136 pages | MR: 1827714 | Zbl: 0993.32009

[11] Kashiwara, Masaki; Schapira, Pierre Microlocal study of ind-sheaves. I. Micro-support and regularity, Astérisque (2003) no. 284, pp. 143-164 (Autour de l’analyse microlocale) | MR: 2003419 | Zbl: 1053.35009

[12] Loday-Richaud, M.; Pourcin, G. On index theorems for linear ordinary differential operators, Ann. Inst. Fourier (Grenoble), Volume 47 (1997) no. 5, pp. 1379-1424 | Article | Numdam | MR: 1600379 | Zbl: 0901.34012

[13] Łojasiewicz, S. Sur le problème de la division, Studia Math., Volume 18 (1959), pp. 87-136 | MR: 107168 | Zbl: 0115.10203

[14] Malgrange, B. Équations différentielles à coefficients polynomiaux, Progress in Mathematics, Volume 96, Birkhäuser Boston Inc., Boston, MA, 1991 | MR: 1117227 | Zbl: 0764.32001

[15] Mochizuki, T. Good formal structure for meromorphic flat connections on smooth projective surfaces (arXiv:math.AG/0803.1346) | Zbl: pre05556812

[16] Mochizuki, T. Wild harmonic bundles and wild pure twistor D-modules (arXiv:math.DG/0803.1344)

[17] Morando, Giovanni An existence theorem for tempered solutions of 𝒟-modules on complex curves, Publ. Res. Inst. Math. Sci., Volume 43 (2007) no. 3, pp. 625-659 | Article | MR: 2361790 | Zbl: 1155.32018

[18] Prelli, Luca Microlocalization of subanalytic sheaves, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 3, pp. 127-132 | MR: 2344810 | Zbl: 1159.14034

[19] Sabbah, Claude Équations différentielles à points singuliers irréguliers en dimension 2, Ann. Inst. Fourier (Grenoble), Volume 43 (1993) no. 5, pp. 1619-1688 | Article | Numdam | MR: 1275212 | Zbl: 0803.32005

[20] Sabbah, Claude Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque (2000) no. 263, viii+190 pages | MR: 1741802 | Zbl: 0947.32005

[21] Sabbah, Claude Déformations isomonodromiques et variétés de Frobenius, Savoirs Actuels (Les Ulis). [Current Scholarship (Les Ulis)], EDP Sciences, Les Ulis, 2002 (, Mathématiques (Les Ulis). [Mathematics (Les Ulis)]) | MR: 1933784

[22] Wasow, Wolfgang Asymptotic expansions for ordinary differential equations, Dover Publications Inc., New York, 1987 (Reprint of the 1976 edition) | MR: 919406 | Zbl: 0644.34003

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