Moduli spaces for linear differential equations and the Painlevé equations

Marius van der Put[1]; Masa-Hiko Saito[2]

  • [1] University of Groningen Institute of Mathematics and Computing Science P.O. Box 407 9700 AK Groningen (The Netherlands)
  • [2] Kobe University Department of Mathematics Graduate School of Science Kobe, Rokko, 657-8501 (Japan)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 7, page 2611-2667
  • ISSN: 0373-0956

Abstract

top
A systematic construction of isomonodromic families of connections of rank two on the Riemann sphere is obtained by considering the analytic Riemann–Hilbert map R H : ℳ → ℛ , where ℳ is a moduli space of connections and ℛ , the monodromy space, is a moduli space for analytic data (i.e., ordinary monodromy, Stokes matrices and links). The assumption that the fibres of R H (i.e., the isomonodromic families) have dimension one, leads to ten moduli spaces ℳ . The induced Painlevé equations are computed explicitly. Except for the Painlevé VI case, these families have irregular singularities. The analytic classification of irregular singularities yields explicit spaces ℛ , which are families of affine cubic surfaces, related to Okamoto–Painlevé pairs. A weak and a strong form of the Riemann–Hilbert problem is treated. Our paper extends the fundamental work of Jimbo–Miwa–Ueno and is related to recent work on Painlevé equations.

How to cite

top

van der Put, Marius, and Saito, Masa-Hiko. "Moduli spaces for linear differential equations and the Painlevé equations." Annales de l’institut Fourier 59.7 (2009): 2611-2667. <http://eudml.org/doc/10466>.

@article{vanderPut2009,
abstract = {A systematic construction of isomonodromic families of connections of rank two on the Riemann sphere is obtained by considering the analytic Riemann–Hilbert map $RH:\mathcal\{M\} \rightarrow \mathcal\{R\}$, where $\mathcal\{M\}$ is a moduli space of connections and $\mathcal\{R\}$, the monodromy space, is a moduli space for analytic data (i.e., ordinary monodromy, Stokes matrices and links). The assumption that the fibres of $RH$ (i.e., the isomonodromic families) have dimension one, leads to ten moduli spaces $\mathcal\{M\}$. The induced Painlevé equations are computed explicitly. Except for the Painlevé VI case, these families have irregular singularities. The analytic classification of irregular singularities yields explicit spaces $\mathcal\{R\}$, which are families of affine cubic surfaces, related to Okamoto–Painlevé pairs. A weak and a strong form of the Riemann–Hilbert problem is treated. Our paper extends the fundamental work of Jimbo–Miwa–Ueno and is related to recent work on Painlevé equations.},
affiliation = {University of Groningen Institute of Mathematics and Computing Science P.O. Box 407 9700 AK Groningen (The Netherlands); Kobe University Department of Mathematics Graduate School of Science Kobe, Rokko, 657-8501 (Japan)},
author = {van der Put, Marius, Saito, Masa-Hiko},
journal = {Annales de l’institut Fourier},
keywords = {Moduli space for linear connections; irregular singularities; Stokes matrices; monodromy spaces; isomonodromic deformations; Painlevé equations; moduli space for linear connections},
language = {eng},
number = {7},
pages = {2611-2667},
publisher = {Association des Annales de l’institut Fourier},
title = {Moduli spaces for linear differential equations and the Painlevé equations},
url = {http://eudml.org/doc/10466},
volume = {59},
year = {2009},
}

TY - JOUR
AU - van der Put, Marius
AU - Saito, Masa-Hiko
TI - Moduli spaces for linear differential equations and the Painlevé equations
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 7
SP - 2611
EP - 2667
AB - A systematic construction of isomonodromic families of connections of rank two on the Riemann sphere is obtained by considering the analytic Riemann–Hilbert map $RH:\mathcal{M} \rightarrow \mathcal{R}$, where $\mathcal{M}$ is a moduli space of connections and $\mathcal{R}$, the monodromy space, is a moduli space for analytic data (i.e., ordinary monodromy, Stokes matrices and links). The assumption that the fibres of $RH$ (i.e., the isomonodromic families) have dimension one, leads to ten moduli spaces $\mathcal{M}$. The induced Painlevé equations are computed explicitly. Except for the Painlevé VI case, these families have irregular singularities. The analytic classification of irregular singularities yields explicit spaces $\mathcal{R}$, which are families of affine cubic surfaces, related to Okamoto–Painlevé pairs. A weak and a strong form of the Riemann–Hilbert problem is treated. Our paper extends the fundamental work of Jimbo–Miwa–Ueno and is related to recent work on Painlevé equations.
LA - eng
KW - Moduli space for linear connections; irregular singularities; Stokes matrices; monodromy spaces; isomonodromic deformations; Painlevé equations; moduli space for linear connections
UR - http://eudml.org/doc/10466
ER -

References

top
  1. D. V. Anosov, A. A. Bolibruch, The Riemann-Hilbert problem, (1994), Friedr. Vieweg & Sohn, Braunschweig Zbl0801.34002MR1276272
  2. Philip Boalch, From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. (3) 90 (2005), 167-208 Zbl1070.34123MR2107041
  3. A. A. Bolibruch, S. Malek, C. Mitschi, On the generalized Riemann-Hilbert problem with irregular singularities, Expo. Math. 24 (2006), 235-272 Zbl1106.34061MR2250948
  4. H. Flaschka, A. C. Newell, Monodromy and spectrum preserving deformations. I, Commun. Math. Phys. 76 (1980), 65-116 Zbl0439.34005MR588248
  5. Robert Fricke, Felix Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, 4 (1965), Johnson Reprint Corp., New York MR183872
  6. Richard Fuchs, Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen, Math. Ann. 63 (1907), 301-321 Zbl38.0362.01MR1511408
  7. B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes, Acta Math. 33 (1910), 1-55 Zbl40.0377.02MR1555055
  8. R. Garnier, Sur les équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale à ses points critiques fixes, Ann. Ecole Norm. Sup. 29 (1912), 1-126 Zbl43.0382.01MR1509146
  9. Masuo Hukuhara, Sur les points singuliers des équations différentielles linéaires. II, Jour. Fac. Soc. Hokkaido Univ. 5 (1937), 123-166 Zbl0016.30502
  10. M. Inaba, Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence, (2006) 
  11. M. Inaba, K. Iwasaki, M.-H. Saito, Dynamics of the sixth Painlevé equation, Théories asymptotiques et équations de Painlevé 14 (2006), 103-167, Soc. Math. France, Paris Zbl1161.34063MR2353464
  12. M. Inaba, K. Iwasaki, M.-H. Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I, Publ. Res. Inst. Math. Sci. 42 (2006), 987-1089 Zbl1127.34055MR2289083
  13. M. Inaba, K. Iwasaki, M.-H. Saito, Moduli of stableparabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II, Adv. Stud. Pure Math. 45 (2006), 387-432, Math. Soc. Japan, Tokyo Zbl1115.14005
  14. K. Iwasaki, An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation, Comm. Math. Phys. 242 (2003), 185-219 Zbl1044.34051MR2018272
  15. Katsunori Iwasaki, A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), 131-135 Zbl1058.34125MR1930217
  16. Michio Jimbo, Tetsuji Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448 Zbl1194.34166MR625446
  17. Michio Jimbo, Tetsuji Miwa, Kimio Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ -function, Phys. D 2 (1981), 306-352 Zbl1194.34167MR630674
  18. B. Malgrange, Sur les déformations isomonodromiques. I. Singularités régulières, Mathematics and physics (Paris, 1979/1982) 37 (1983), 401-426, Birkhäuser Boston, Boston, MA Zbl0528.32017MR728431
  19. Bernard Malgrange, Déformations isomonodromiques, forme de Liouville, fonction τ , Ann. Inst. Fourier (Grenoble) 54 (2004), 1371-1392, xiv, xx Zbl1086.34071MR2127851
  20. Yousuke Ohyama, Hiroyuki Kawamuko, Hidetaka Sakai, Kazuo Okamoto, Studies on the Painlevé equations. V. Third Painlevé equations of special type P III ( D 7 ) and P III ( D 8 ) , J. Math. Sci. Univ. Tokyo 13 (2006), 145-204 Zbl1170.34061MR2277519
  21. Yousuke Ohyama, Shoji Okumura, A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations, J. Phys. A 39 (2006), 12129-12151 Zbl1116.34072MR2266216
  22. Kazuo Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math. (N.S.) 5 (1979), 1-79 Zbl0426.58017MR614694
  23. Paul Painlevé, Oeuvres de Paul Painlevé. Tome I, (1973), Éditions du Centre National de la Recherche Scientifique, Paris Zbl1092.01510MR532682
  24. Marius van der Put, Michael F. Singer, Galois theory of linear differential equations, 328 (2003), Springer-Verlag, Berlin Zbl1036.12008MR1960772
  25. Masa-Hiko Saito, Taro Takebe, Hitomi Terajima, Deformation of Okamoto-Painlevé pairs and Painlevé equations, J. Algebraic Geom. 11 (2002), 311-362 Zbl1022.34079MR1874117
  26. Masa-Hiko Saito, Hitomi Terajima, Nodal curves and Riccati solutions of Painlevé equations, J. Math. Kyoto Univ. 44 (2004), 529-568 Zbl1117.14015MR2103782
  27. H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229 Zbl1010.34083MR1882403
  28. Y. Shibuya, Perturbation of linear ordinary differential equations at irregular singular points, Funkcial. Ekvac. 11 (1968), 235-246 Zbl0228.34036MR243171
  29. H. Terajima, Families of Okamoto-Painlevé pairs and Painlevé equations, Ann. Mat. Pura Appl. (4) 186 (2007), 99-146 Zbl1232.34120MR2263893
  30. H. L. Turrittin, Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta Math. 93 (1955), 27-66 Zbl0064.33603MR68689

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.