Kyoto Journal of Mathematics

Moduli of unramified irregular singular parabolic connections on a smooth projective curve

Abstract

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover, we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links, and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann–Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit the geometric Painlevé property as in the regular singular cases proved generally.

Article information

Source
Kyoto J. Math., Volume 53, Number 2 (2013), 433-482.

Dates
First available in Project Euclid: 20 May 2013

https://projecteuclid.org/euclid.kjm/1369071235

Digital Object Identifier
doi:10.1215/21562261-2081261

Mathematical Reviews number (MathSciNet)
MR3079310

Zentralblatt MATH identifier
1267.14015

Citation

Inaba, Michi-aki; Saito, Masa-Hiko. Moduli of unramified irregular singular parabolic connections on a smooth projective curve. Kyoto J. Math. 53 (2013), no. 2, 433--482. doi:10.1215/21562261-2081261. https://projecteuclid.org/euclid.kjm/1369071235

References

• [1] D. G. Babbitt and V. S. Varadarajan. Local Moduli for Meromorphic Differential Equations, Astérisque 169-170, Soc. Math. France, Montrouge, 1989.
• [2] C. L. Bermer and D. S. Sage, Moduli spaces of irregular singular connections, preprint, arXiv:1004.4411v2 [math.AG].
• [3] O. Biquard and P. Boalch, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179–204.
• [4] P. Boalch, Symplectic manifolds and isomonodromic deformations. Adv. Math. 163 (2001), 137–205.
• [5] A. A Bolibruch, S. Malek, and C. Mitschi, On the generalized Riemann–Hilbert problem with irregular singularities, Expo. Math. 24 (2006), 235–272.
• [6] B. Gambier, Sur les équations différentiells du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes, Acta Math. 33 (1910), 1–55.
• [7] T. Hausel and F. Rodriguez-Villegas, Mixed Hodge polynomials of character varieties, with an appendix by Nicholas M. Katz, Invent. Math. 174 (2008), 555-624.
• [8] M. Inaba, Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence, to appear in J. Algebraic Geom., preprint, arXiv:0602004v2 [math.AG].
• [9] M. Inaba, K. Iwasaki, and M.-H. Saito, “Dynamics of the sixth Pailevé equations” in Théories asymptotiques et équations de Painlevé, Sémin. Congr. 14, Soc. Math. France, Paris, 2006, 103–167.
• [10] M. Inaba, K. Iwasaki, and 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.
• [11] M. Inaba, K. Iwasaki, and M.-H. Saito, “Moduli of stable parabolic connections, Riemann–Hilbert correspondence and geometry of Painlevé equation of type VI, II” in Moduli Spaces and Arithmetic Geometry, Adv. Stud. Pure Math. 45, Math. Soc. Japan, Tokyo, 2006, 387–432.
• [12] M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, II, Physica D 2 (1981), 407–448.
• [13] M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, I: General theory and $\tau$-function, Physica D 2 (1981), 306–352.
• [14] B. Malgrange, “Sur les déformations isomonodromiques, I: Singularités régulières” in Mathematics and Physics (Paris, 1979/1982), Progr. Math. 37, Birkhäuser, Boston, 401–426.
• [15] B. Malgrange, “Sur les déformations isomonodromiques, II: Singularités irrégulières” in Mathematics and Physics (Paris, 1979/1982), Progr. Math. 37, Birkhäuser, Boston, 1983, 427–438.
• [16] T. Miwa, Painlevé property of monodromy preserving equations and the analyticity of $\tau$ functions, Publ. Res. Inst. Math. Sci. 17 (1981), 709–721.
• [17] K. Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Espaces des conditions initiales, Japan. J. Math. (N.S.) 5 (1979), 1–79.
• [18] P. Painlevé, Mémoire sur les équations différentielles dont l’integrale generale est uniforme, Bull. Soc. Math. France 28, (1900), 206–261.
• [19] P. Painlevé, Sur les équations différentielles du second ordre à points critiques fixes, C. R. Acad. Sci. Paris 143 (1906), 1111–1117.
• [20] M. van der Put and M.-H. Saito, Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble) 59 (2009), 2611–2667.
• [21] M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren Math. Wiss. 328, Springer, Berlin, 2003.
• [22] C. Sabbah, Isomonodromic Deformations and Frobenius Manifolds: An Introduction, Universitext, Springer, London; EDP Sciences, Les Ulis, France, 2007.
• [23] M.-H. Saito and T. Takebe, Classification of Okamoto–Painlevé Pairs, Kobe J. Math. 19 (2002), 21–55.
• [24] M.-H. Saito, T. Takebe, and H. Terajima, Deformation of Okamoto–Painlevé pairs and Painlevé equations, J. Algebraic Geom. 11 (2002), 311–362.
• [25] H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165–229.
• [26] Y. Sibuya, Perturbation of linear ordinary differential equations at irregular singular points, Funkcial. Ekvac. 11 (1968), 235–246.
• [27] Y. Sibuya, Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation, Transl. Math. Monogr. 82, Amer. Math. Soc., Providence, 1990.