Table of contents for this issue | Previous article | Next article
Giovanni Morando
Tempered solutions of $\mathcal{D}$-modules on complex curves and formal invariants
(Solutions holomorphes tempérées des $\mathcal{D}$-modules sur les courbes complexes et invariants formels)
Annales de l'institut Fourier, 59 no. 4 (2009), p. 1611-1639, doi: 10.5802/aif.2472
Article: subscription required (your ip address: 107.21.156.140) | Reviews MR 2566969 | Zbl pre05614567
Class. Math.: 34M35, 32B20, 34Mxx
Keywords: $\mathcal{D}$-modules, irregular singularities, tempered holomorphic functions, subanalytic
[1] D. G. Babbitt & V. S. Varadarajan, “Local moduli for meromorphic differential equations”, Astérisque (1989) no. 169-170 MR 1014083 | Zbl 0683.34003
[2] Edward Bierstone & Pierre D. Milman, “Semianalytic and subanalytic sets”, Inst. Hautes Études Sci. Publ. Math. (1988) no. 67, p. 5-42
Numdam | MR 972342 | Zbl 0674.32002
[3] Jacek Bochnak, Michel Coste & Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 36, Springer-Verlag, 1998, Translated from the 1987 French original, Revised by the authors MR 1659509 | Zbl 0912.14023
[4] Alain Chenciner, 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, 1978 Zbl 0557.14016
[5] Michel Coste, An introduction to o-minimal geometry, Istituti Ed. e Poligrafici Intern., Università di Pisa, Dipartimento di Matematica, 2000
[6] Pierre Deligne, Bernard Malgrange & Jean-Pierre Ramis, Singularités irrégulières, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 5, Société Mathématique de France, 2007, Correspondance et documents. [Correspondence and documents] MR 2387754 | Zbl 1130.14001
[7] M. Kashiwara, Faisceaux constructibles et systèmes holonômes d’équations aux dérivées partielles linéaires à points singuliers réguliers, Séminaire Goulaouic-Schwartz, 1979–1980 (French), École Polytechnique, 1980, p. 7
Cedram | MR 600704 | Zbl 0444.58014
[8] Masaki Kashiwara, “The Riemann-Hilbert problem for holonomic systems”, Publ. Res. Inst. Math. Sci. 20 (1984) no. 2, p. 319-365
Article | MR 743382 | Zbl 0566.32023
[9] Masaki Kashiwara, $D$-modules and microlocal calculus, Translations of Mathematical Monographs 217, American Mathematical Society, 2003, Translated from the 2000 Japanese original by Mutsumi Saito, Iwanami Series in Modern Mathematics MR 1943036 | Zbl 1017.32012
[10] Masaki Kashiwara & Pierre Schapira, “Ind-sheaves”, Astérisque (2001) no. 271 MR 1827714 | Zbl 0993.32009
[11] Masaki Kashiwara & Pierre Schapira, “Microlocal study of ind-sheaves. I. Micro-support and regularity”, Astérisque (2003) no. 284, p. 143-164, Autour de l’analyse microlocale MR 2003419 | Zbl 1053.35009
[12] M. Loday-Richaud & G. Pourcin, “On index theorems for linear ordinary differential operators”, Ann. Inst. Fourier (Grenoble) 47 (1997) no. 5, p. 1379-1424
Cedram | MR 1600379 | Zbl 0901.34012
[13] S. Łojasiewicz, “Sur le problème de la division”, Studia Math. 18 (1959), p. 87-136
Article | MR 107168 | Zbl 0115.10203
[14] B. Malgrange, Équations différentielles à coefficients polynomiaux, Progress in Mathematics 96, Birkhäuser Boston Inc., 1991 MR 1117227 | Zbl 0764.32001
[15] T. Mochizuki, “Good formal structure for meromorphic flat connections on smooth projective surfaces”, arXiv:math.AG/0803.1346
arXiv | Zbl pre05556812
[16] T. Mochizuki, “Wild harmonic bundles and wild pure twistor D-modules”, arXiv:math.DG/0803.1344
arXiv
[17] Giovanni Morando, “An existence theorem for tempered solutions of $\mathcal{D}$-modules on complex curves”, Publ. Res. Inst. Math. Sci. 43 (2007) no. 3, p. 625-659
Article | MR 2361790 | Zbl 1155.32018
[18] Luca Prelli, “Microlocalization of subanalytic sheaves”, C. R. Math. Acad. Sci. Paris 345 (2007) no. 3, p. 127-132 MR 2344810 | Zbl 1159.14034
[19] Claude Sabbah, “Équations différentielles à points singuliers irréguliers en dimension $2$”, Ann. Inst. Fourier (Grenoble) 43 (1993) no. 5, p. 1619-1688
Cedram | MR 1275212 | Zbl 0803.32005
[20] Claude Sabbah, “Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2”, Astérisque (2000) no. 263 MR 1741802 | Zbl 0947.32005
[21] Claude Sabbah, 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] Wolfgang Wasow, Asymptotic expansions for ordinary differential equations, Dover Publications Inc., 1987, Reprint of the 1976 edition MR 919406 | Zbl 0644.34003
Giovanni Morando
Tempered solutions of $\mathcal{D}$-modules on complex curves and formal invariants
(Solutions holomorphes tempérées des $\mathcal{D}$-modules sur les courbes complexes et invariants formels)
Annales de l'institut Fourier, 59 no. 4 (2009), p. 1611-1639, doi: 10.5802/aif.2472
Article: subscription required (your ip address: 107.21.156.140) | Reviews MR 2566969 | Zbl pre05614567
Class. Math.: 34M35, 32B20, 34Mxx
Keywords: $\mathcal{D}$-modules, irregular singularities, tempered holomorphic functions, subanalytic
Résumé - Abstract
Let $X$ be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of $\mathcal{D}$-modules on $X$ induces a fully faithful functor on a subcategory of germs of formal holonomic $\mathcal{D}$-modules. Further, given a germ $\mathcal{M}$ of holonomic $\mathcal{D}$-module, we obtain some results linking the subanalytic sheaf of tempered solutions of $\mathcal{M}$ and the classical formal and analytic invariants of $\mathcal{M}$.
Bibliography
[2] Edward Bierstone & Pierre D. Milman, “Semianalytic and subanalytic sets”, Inst. Hautes Études Sci. Publ. Math. (1988) no. 67, p. 5-42
Numdam | MR 972342 | Zbl 0674.32002
[3] Jacek Bochnak, Michel Coste & Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 36, Springer-Verlag, 1998, Translated from the 1987 French original, Revised by the authors MR 1659509 | Zbl 0912.14023
[4] Alain Chenciner, 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, 1978 Zbl 0557.14016
[5] Michel Coste, An introduction to o-minimal geometry, Istituti Ed. e Poligrafici Intern., Università di Pisa, Dipartimento di Matematica, 2000
[6] Pierre Deligne, Bernard Malgrange & Jean-Pierre Ramis, Singularités irrégulières, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 5, Société Mathématique de France, 2007, Correspondance et documents. [Correspondence and documents] MR 2387754 | Zbl 1130.14001
[7] M. Kashiwara, Faisceaux constructibles et systèmes holonômes d’équations aux dérivées partielles linéaires à points singuliers réguliers, Séminaire Goulaouic-Schwartz, 1979–1980 (French), École Polytechnique, 1980, p. 7
Cedram | MR 600704 | Zbl 0444.58014
[8] Masaki Kashiwara, “The Riemann-Hilbert problem for holonomic systems”, Publ. Res. Inst. Math. Sci. 20 (1984) no. 2, p. 319-365
Article | MR 743382 | Zbl 0566.32023
[9] Masaki Kashiwara, $D$-modules and microlocal calculus, Translations of Mathematical Monographs 217, American Mathematical Society, 2003, Translated from the 2000 Japanese original by Mutsumi Saito, Iwanami Series in Modern Mathematics MR 1943036 | Zbl 1017.32012
[10] Masaki Kashiwara & Pierre Schapira, “Ind-sheaves”, Astérisque (2001) no. 271 MR 1827714 | Zbl 0993.32009
[11] Masaki Kashiwara & Pierre Schapira, “Microlocal study of ind-sheaves. I. Micro-support and regularity”, Astérisque (2003) no. 284, p. 143-164, Autour de l’analyse microlocale MR 2003419 | Zbl 1053.35009
[12] M. Loday-Richaud & G. Pourcin, “On index theorems for linear ordinary differential operators”, Ann. Inst. Fourier (Grenoble) 47 (1997) no. 5, p. 1379-1424
Cedram | MR 1600379 | Zbl 0901.34012
[13] S. Łojasiewicz, “Sur le problème de la division”, Studia Math. 18 (1959), p. 87-136
Article | MR 107168 | Zbl 0115.10203
[14] B. Malgrange, Équations différentielles à coefficients polynomiaux, Progress in Mathematics 96, Birkhäuser Boston Inc., 1991 MR 1117227 | Zbl 0764.32001
[15] T. Mochizuki, “Good formal structure for meromorphic flat connections on smooth projective surfaces”, arXiv:math.AG/0803.1346
arXiv | Zbl pre05556812
[16] T. Mochizuki, “Wild harmonic bundles and wild pure twistor D-modules”, arXiv:math.DG/0803.1344
arXiv
[17] Giovanni Morando, “An existence theorem for tempered solutions of $\mathcal{D}$-modules on complex curves”, Publ. Res. Inst. Math. Sci. 43 (2007) no. 3, p. 625-659
Article | MR 2361790 | Zbl 1155.32018
[18] Luca Prelli, “Microlocalization of subanalytic sheaves”, C. R. Math. Acad. Sci. Paris 345 (2007) no. 3, p. 127-132 MR 2344810 | Zbl 1159.14034
[19] Claude Sabbah, “Équations différentielles à points singuliers irréguliers en dimension $2$”, Ann. Inst. Fourier (Grenoble) 43 (1993) no. 5, p. 1619-1688
Cedram | MR 1275212 | Zbl 0803.32005
[20] Claude Sabbah, “Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2”, Astérisque (2000) no. 263 MR 1741802 | Zbl 0947.32005
[21] Claude Sabbah, 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] Wolfgang Wasow, Asymptotic expansions for ordinary differential equations, Dover Publications Inc., 1987, Reprint of the 1976 edition MR 919406 | Zbl 0644.34003
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310