Nous considérons les fibrés à connexion méromorphe sans trace de rang 2 sur les surfaces de Riemann compactes de genre quelconque. En déformant la courbe, la position des pôles et le fibré à connexion, nous construisons la déformation isomonodromique universelle globale d’un tel fibré à connexion initial. Notre construction spécifique au cas du rang 2 et sans trace est plus élémentaire que la construction en rang quelconque due à B. Malgrange et I. Krichever au sens où elle ne nécessite pas d’analyse de Stokes des singularités irrégulières. De plus, elle englobe le cas des singularités résonantes de manière naturelle.
We consider tracefree meromorphic rank 2 connections over compact Riemann surfaces of arbitrary genus. By deforming the curve, the position of the poles and the connection, we construct the global universal isomonodromic deformation of such a connection. Our construction, which is specific to the tracefree rank 2 case, does not need any Stokes analysis for irregular singularities. It is thereby more elementary than the construction in arbitrary rank due to B. Malgrange and I. Krichever and it includes the case of resonant singularities in a natural way.
@article{AIF_2010__60_2_515_0,
author = {Heu, Viktoria},
title = {Universal isomonodromic deformations of meromorphic rank 2 connections on curves},
journal = {Annales de l'Institut Fourier},
volume = {60},
year = {2010},
pages = {515-549},
doi = {10.5802/aif.2531},
zbl = {1193.32009},
mrnumber = {2667785},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2010__60_2_515_0}
}
Heu, Viktoria. Universal isomonodromic deformations of meromorphic rank 2 connections on curves. Annales de l'Institut Fourier, Tome 60 (2010) pp. 515-549. doi : 10.5802/aif.2531. http://gdmltest.u-ga.fr/item/AIF_2010__60_2_515_0/
[1] Extensions de Deligne pour les croisements normaux, Éléments de la théorie des systèmes différentiels géométriques, Soc. Math. France, Paris (Sémin. Congr.) Tome 8 (2004), pp. 149-164 | MR 2077648 | Zbl 1070.32006
[2] Équations différentielles à points singuliers réguliers, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 163 (1970) | MR 417174 | Zbl 0244.14004
[3] Poisson structures and their normal forms, Birkhäuser Verlag, Basel, Progress in Mathematics, Tome 242 (2005) | MR 2178041 | Zbl 1082.53078
[4] Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, Tome 1965 (1965), pp. 89-94 | MR 184258 | Zbl 0135.12601
[5] Déformations isomonodromiques des connexions de rang 2 sur les courbes, Université Rennes 1 (2008) (Ph. D. Thesis)
[6] Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY (2006) | MR 2245223 | Zbl 1102.30001
[7] An overview of Deligne’s work on Hilbert’s twenty-first problem, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974), Amer. Math. Soc., Providence, R. I. (1976), pp. 537-557 | MR 432640 | Zbl 0347.14010
[8] Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations, Mosc. Math. J., Tome 2 (2002) no. 4, p. 717-752, 806 | MR 1986088 | Zbl 1044.70010
[9] Transversely projective foliations on surfaces: existence of minimal form and prescription of monodromy, Internat. J. Math., Tome 18 (2007) no. 6, pp. 723-747 | Article | MR 2337401 | Zbl 1124.37028
[10] Déformations de systèmes différentiels et microdifférentiels, Mathematics and physics (Paris, 1979/1982), Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 37 (1983), pp. 353-379 | MR 728429 | Zbl 0528.32016
[11] Sur les déformations isomonodromiques. I. Singularités régulières, Mathematics and physics (Paris, 1979/1982), Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 37 (1983), pp. 401-426 | MR 728431 | Zbl 0528.32017
[12] Sur les déformations isomonodromiques. II. Singularités irrégulières, Mathematics and physics (Paris, 1979/1982), Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 37 (1983), pp. 427-438 | MR 728432 | Zbl 0528.32018
[13] Deformations of differential systems. II, J. Ramanujan Math. Soc., Tome 1 (1986) no. 1-2, pp. 3-15 | MR 945599 | Zbl 0687.32019
[14] Connexions méromorphes 2 Le réseau canonique, Invent. math. (1996) no. 124, pp. 367-387 | Article | MR 1369422 | Zbl 0849.32003
[15] Déformations isomonodromiques, forme de Liouville, fonction , Ann. Inst. Fourier (Grenoble), Tome 54 (2004) no. 5, p. 1371-1392, xiv, xx | Article | Numdam | MR 2127851 | Zbl 1086.34071
[16] Modules de feuilletages holomorphes singuliers. I. Équisingularité, Invent. Math., Tome 103 (1991) no. 2, pp. 297-325 | Article | MR 1085109 | Zbl 0709.32025
[17] Equisingular unfoldings of foliations by curves, Astérisque (1994) no. 222, pp. 6, 285-302 (Complex analytic methods in dynamical systems (Rio de Janeiro, 1992)) | MR 1285392 | Zbl 0826.32026
[18] On the volume elements on a manifold, Trans. Amer. Math. Soc., Tome 120 (1965), pp. 286-294 | Article | MR 182927 | Zbl 0141.19407
[19] The complex analytic theory of Teichmüller spaces, John Wiley & Sons Inc., New York, Canadian Mathematical Society Series of Monographs and Advanced Texts (1988) (A Wiley-Interscience Publication) | MR 927291 | Zbl 0667.30040
[20] Isomonodromic deformation and Painlevé equations, and the Garnier system, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 33 (1986) no. 3, pp. 575-618 | MR 866050 | Zbl 0631.34011
[21] Zeros of the Jimbo, Miwa, Ueno tau function, J. Math. Phys., Tome 40 (1999) no. 12, pp. 6638-6681 | Article | MR 1725878 | Zbl 0974.34081
[22] Linear meromorphic differential equations: a modern point of view, Bull. Amer. Math. Soc. (N.S.), Tome 33 (1996) no. 1, pp. 1-42 | Article | MR 1339809 | Zbl 0862.34004