Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T11:19:38.722Z Has data issue: false hasContentIssue false
  • English
  • Français

The Stokes Structure of a good meromorphic flat bundle

Part of: Partial differential equations Analytic spaces

Published online by Cambridge University Press:  03 May 2011

Takuro Mochizuki
Takuro Mochizuki
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan (takuro@kurims.kyoto-u.ac.jp)
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a survey on the Stokes structure of a good meromorphic flat bundle. We also show that a meromorphic flat bundle has the good formal structure if and only if it has a good lattice.

Keywords

meromorphic flat bundleStokes structureRiemann–Hilbert–Birkhoff correspondence

MSC classification

Secondary: 32C38: Sheaves of differential operators and their modules, $D$-modules 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Special Issue 3: A Special Issue in Honour of Louis Boutet de Monvel and Pierre Schapira , July 2011 , pp. 675 - 712
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Balser, W., Jurkat, W. B. and Lutz, D. A., Birkhoff invariants and Stokes' multipliers for meromorphic linear differential equations, J. Math. Analysis Applic. 71 (1979), 48–94.CrossRefGoogle Scholar
2.Banica, C., Le complété formel d'un espace analytique le long d'un sous-espace: un théoréme de comparaison, Manuscr. Math. 6 (1972), 207–244.CrossRefGoogle Scholar
3.Bingener, J., Über Formale Komplexe Räume, Manuscr. Math. 24 (1978), 253–293.CrossRefGoogle Scholar
4.Deligne, P., Équation différentielles à points singuliers reguliers, Lectures Notes in Mathematics, Volume 163 (Springer, 1970).CrossRefGoogle Scholar
5.Deligne, P., Malgrange, B. and Ramis, J.-P., Singularités irrégulières, Documents Mathématiques, Volume 5 (Société Mathématique de France, Paris, 2007).Google Scholar
6.Hartshorne, R., On the De Rham cohomology of algebraic varieties, Publ. Math. IHES 45 (1975), 5–99.CrossRefGoogle Scholar
7.Jimbo, M., Miwa, T. and Ueno, K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, I, General theory and τ-function, Physica D 2 (1981), 306–352.CrossRefGoogle Scholar
8.Kashiwara, M., Regular holonomic D-modules and distributions on complex manifolds, Adv. Stud. Pure Math. 8 (1987), 199–206.CrossRefGoogle Scholar
9.Kedlaya, K., Good formal structures for flat meromorphic connections, II, Excellent schemes, J. Am. Math. Soc. 24 (2011), 183–229.CrossRefGoogle Scholar
10.Kedlaya, K., Good formal structures for flat meromorphic connections, I, Surfaces, Duke Math. J. 154 (2010), 343–418.CrossRefGoogle Scholar
11.Levelt, A., Jordan decomosition for a class of singular differential operators, Arkiv Mat. 13 (1975), 1–27CrossRefGoogle Scholar
12.Majima, H., Asymptotic analysis for integrable connections with irregular singular points, Lecture Notes in Mathematics, Volume 1075 (Springer, 1984).CrossRefGoogle Scholar
13.Malgrange, B., Déformations de systèmes différentiels et microdifférentiels, in Mathematics and physics (Paris, 1979/1982), Progress in Mathematics, Volume 37, pp. 353–379 (Birkhäuser, Boston, MA, 1983).Google Scholar
14.Malgrange, B., La classification des connexions irrégulières à une variable, in Mathematics and physics (Paris, 1979/1982), Progress in Mathematics, Volume 37, pp. 381–399 (Birkhäuser, Boston, MA, 1983).Google Scholar
15.Malgrange, B., Équations différentielles à coefficients polynomiaux, Progress in Mathematics, Volume 96 (Birkhäuser, Boston, MA, 1991).Google Scholar
16.Malgrange, B., Connexions méromorphies 2, Le réseau canonique, Invent. Math. 124 (1996), 367–387.CrossRefGoogle Scholar
17.Mebkhout, Z., Le théorème de comparaison entre cohomologies de de Rham sur le corps des nombres complexes, C. R. Acad. Sci. Paris Sér. I 305 (1987), 549–552.Google Scholar
18.Mebkhout, Z., Le théorème de comparaison entre cohomologies de de Rham d'une variété algèbrique complexe et le théorème d'existence de Riemann, Publ. Math. IHES 69 (1989), 47–89.CrossRefGoogle Scholar
19.Mochizuki, T., Good formal structure for meromorphic flat connections on smooth projective surfaces, in Algebraic analysis and around, Advanced Studies in Pure Mathematics, Volume 54 (American Mathematical Society, Providence, RI, 2009).Google Scholar
20.Mochizuki, T., On Deligne–Malgrange lattices, resolution of turning points and harmonic bundles, Annales Inst. Fourier 59 (2009), 2819–2837.CrossRefGoogle Scholar
21.Mochizuki, T., Wild harmonic bundles and wild pure twistor D-modules, Astérisque, in press.Google Scholar
22.Mochizuki, T., Asymptotic behaviour of variation of pure polarized TERP structures, Publ. RIMS Kyoto, in press.Google Scholar
23.Sabbah, C., Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2, Astérisque, Volume 263 (Société Mathématique de France, Paris, 2000).Google Scholar
24.Sabbah, C., Introduction to Stokes structures, preprint (arXiv:0912.2762).Google Scholar
25.Sibuya, Y., Linear differential equations in the complex domain: problems of analytic continuation (Kinokuniya, Tokyo, 1976) (in Japanese; translation appeared in Translations of Mathematical Monographs, Volume 82 (American Mathematical Society, Providence, RI, 1990)).Google Scholar
26.van der Put, M. and Saito, M.-H., Moduli spaces for linear differential equations and the Painlevé equations, Annales Inst. Fourier 59 (2009), 2611–2667.CrossRefGoogle Scholar