Tempered solutions of 𝒟-modules on complex curves and formal invariants
[Solutions holomorphes tempérées des 𝒟-modules sur les courbes complexes et invariants formels]
Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1611-1639.

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 de 𝒟-module holonome, nous obtenons des résultats qui lient le faisceau sous-analytique des solutions tempérées de avec les invariants formels et analytiques classiques de .

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 of holonomic 𝒟-module, we obtain some results linking the subanalytic sheaf of tempered solutions of and the classical formal and analytic invariants of .

DOI : 10.5802/aif.2472
Classification : 34M35, 32B20, 34Mxx
Keywords: $\mathcal{D}$-modules, irregular singularities, tempered holomorphic functions, subanalytic
Mot clés : $\mathcal{D}$-modules, singularités irréguliers, fonctions holomorphes tempérées, sous-analytique
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, Tome 59 (2009) no. 4, pp. 1611-1639. doi : 10.5802/aif.2472. http://www.numdam.org/articles/10.5802/aif.2472/

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

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

[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)], 36, Springer-Verlag, Berlin, 1998 (Translated from the 1987 French original, Revised by the authors) | MR | Zbl

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

[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 | Zbl

[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, pp. 7 (Exp. No. 19) | Numdam | MR | Zbl

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

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

[10] Kashiwara, Masaki; Schapira, Pierre Ind-sheaves, Astérisque (2001) no. 271, pp. 136 | Numdam | MR | Zbl

[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) | Numdam | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

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

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

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

[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

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

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