On Deligne-Malgrange lattices, resolution of turning points and harmonic bundles
[Sur réseaux de Deligne-Malgrange, résolution des tournants, et fibrés harmoniques]
Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 2819-2837.

Nous donnons un sommaire du développement récent de l’étude sur les réseaux de Deligne-Malgrange et la résolution des tournants pour les connexions méromorphes plates algébriques. Nous expliquons également leur relation avec les fibrés harmoniques sauvages. L’auteur espère que ce sera utile pour l’accès à son travail sur les fibrés harmoniques sauvages.

In this short survey, we would like to overview the recent development of the study on Deligne-Malgrange lattices and resolution of turning points for algebraic meromorphic flat bundles. We also explain their relation with wild harmonic bundles. The author hopes that it would be helpful for access to his work on wild harmonic bundles.

DOI : 10.5802/aif.2509
Classification : 14J60, 32C38, 53C07
Keywords: Harmonic bundle, meromorphic flat bundle, Deligne-Malgrange lattice
Mot clés : fibrés harmoniques, connexions méromorphies plates, réseau de Deligne-Malgrange
Mochizuki, Takuro 1

1 Kyoto University Research Institute for Mathematical Sciences Kyoto 606-8502 (Japan)
@article{AIF_2009__59_7_2819_0,
     author = {Mochizuki, Takuro},
     title = {On {Deligne-Malgrange} lattices, resolution of turning points and harmonic bundles},
     journal = {Annales de l'Institut Fourier},
     pages = {2819--2837},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {7},
     year = {2009},
     doi = {10.5802/aif.2509},
     mrnumber = {2649340},
     zbl = {1202.32008},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2509/}
}
TY  - JOUR
AU  - Mochizuki, Takuro
TI  - On Deligne-Malgrange lattices, resolution of turning points and harmonic bundles
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 2819
EP  - 2837
VL  - 59
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2509/
DO  - 10.5802/aif.2509
LA  - en
ID  - AIF_2009__59_7_2819_0
ER  - 
%0 Journal Article
%A Mochizuki, Takuro
%T On Deligne-Malgrange lattices, resolution of turning points and harmonic bundles
%J Annales de l'Institut Fourier
%D 2009
%P 2819-2837
%V 59
%N 7
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2509/
%R 10.5802/aif.2509
%G en
%F AIF_2009__59_7_2819_0
Mochizuki, Takuro. On Deligne-Malgrange lattices, resolution of turning points and harmonic bundles. Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 2819-2837. doi : 10.5802/aif.2509. http://archive.numdam.org/articles/10.5802/aif.2509/

[1] André, Y. Structure des connexions méromorphes formelles de plusieurs variables et semi-continuité de l’irrégularité, Invent. Math., Volume 170 (2007), pp. 147-198 | DOI | MR | Zbl

[2] Bingener, J. Über Formale Komplexe Räume, Manuscripta Math., Volume 24 (1978), pp. 253-293 | DOI | MR | Zbl

[3] Biquard, O.; Boalch, P. Wild non-abelian Hodge theory on curves, 140, Compos. Math., 2004 | MR | Zbl

[4] Corlette, K. Flat G-bundles with canonical metrics, J. Differential Geom., Volume 28 (1988), pp. 361-382 | MR | Zbl

[5] Deligne, P. Équation différentielles à points singuliers réguliers, Lectures Notes in Maths., Volume 163, Springer, 1970 | MR | Zbl

[6] Jost, J.; Zuo, K. Harmonic maps of infinite energy and rigidity results for representations of fundamental groups of quasiprojective varieties, J. Differential Geom., Volume 47 (1997), pp. 469-503 | MR | Zbl

[7] Katz, N. Nilpotent connections and the monodromy theorem; applications of a result of Turrittin, Publ. Math. I.H.E.S., Volume 39 (1970), pp. 175-232 | Numdam | MR | Zbl

[8] Kedlaya, K. Good formal structures for flat meromorphic connections, I; Surfaces (math:0811.0190)

[9] Laszlo, Y.; Pauly, C. On the Hitchin morphism in positive characteristic, Intenat. Math. Res. Notices, 2001 (129–143, math.AG/0005044) | MR | Zbl

[10] Levelt, A. Jordan decomosition for a class of singular differential operators, Ark. Math., Volume 13 (1975), pp. 1-27 | DOI | MR | Zbl

[11] Majima, H. Asymptotic analysis for integrable connections with irregular singular points, Lecture Notes in Mathematics, Volume 1075, Springer-Verlag, Berlin, 1984 | MR | Zbl

[12] Malgrange, B. Connexions méromorphies 2, Le réseau canonique, Invent. Math., Volume 124 (1996), pp. 367-387 | DOI | MR | Zbl

[13] Mochizuki, T. Asymptotic behaviour of variation of pure polarized TERP structures (math:0811.1384, Version 1)

[14] Mochizuki, T. Wild harmonic bundles and wild pure twistor D-modules (math:0803.1344, Version 3)

[15] Mochizuki, T. Kobayashi-Hitchin correspondence for tame harmoinc bundles and an application, Astérisque, Volume 309, Société Mathématique de France, 2006 | Numdam | MR | Zbl

[16] Mochizuki, T. Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D -modules, I, II, 185, Mem. Amer. Math. Soc., 2007 | MR

[17] Mochizuki, T. Good formal structure for meromorphic flat connections on smooth projective surfaces, Algebraic Analysis and Around, Volume 54, Advanced Studies in Pure Mathematics, 2009, pp. 223-253 (math:0803.1346) | MR

[18] Mochizuki, T. Kobayashi-Hitchin correspondence for tame harmonic bundles II, Geometry & Topology, Volume 13 (2009), pp. 359-455 (math.DG/0602266) | DOI | MR | Zbl

[19] Sabbah, C. Harmonic metrics and connections with irregular singularities, Ann. Inst. Fourier (Grenoble), Volume 49 (1999), pp. 1265-1291 | DOI | Numdam | MR | Zbl

[20] Sabbah, C. Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque, Volume 263, Société Mathématique de France, Paris, 2000 | MR | Zbl

[21] Simpson, C. Constructing variations of Hodge structure using Yang-Mills theory and application to uniformization, J. Amer. Math. Soc., Volume 1 (1988), pp. 867-918 | DOI | MR | Zbl

[22] Simpson, C. Harmonic bundles on non-compact curves, J. Amer. Math. Soc., Volume 3 (1990), pp. 713-770 | DOI | MR | Zbl

[23] Wasow, W. Asymptotic expansions for ordinary equations, 1987 (Reprint of 1976 edition. Dover Publications, Inc., New York) | Zbl

Cité par Sources :