Abstract
We provide a unique normal form for rank two irregular connections on the Riemann sphere. In fact, we provide a birational model where we introduce apparent singular points and where the bundle has a fixed Birkhoff–Grothendieck decomposition. The essential poles and the apparent poles provide two parabolic structures. The first one only depends on the formal type of the singular points. The latter one determines the connection (accessory parameters). As a consequence, an open set of the corresponding moduli space of connections is canonically identified with an open set of some Hilbert scheme of points on the explicit blow-up of some Hirzebruch surface. This generalizes previous results obtained by Szabó to the irregular case. Our work is more generally related to ideas and descriptions of Oblezin, Dubrovin–Mazzocco, and Saito–Szabó in the logarithmic case. After the first version of this work appeared, Komyo used our normal form to compute isomonodromic Hamiltonian systems for irregular Garnier systems.
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Notes
It has been recently generalized to higher rank and genus in the logarithmic setting by Saito and Szabó in [22].
Up to a sign, depending on authors’ conventions.
From now on, we omit the subscript \({{\mathbb {P}}^1}\) in the line bundles for notational simplicity.
References
D. Arinkin, R. Fedorov, An example of the Langlands correspondence for irregular rank two connections on \(\mathbb{P}^1\). Adv. Math. 230, 1078–1123 (2012)
P. Boalch, Symplectic manifolds and isomonodromic deformations. Adv. Math. 163, 137–205 (2001)
A.A. Bolibruch, S. Malek, C. Mitschi, On the generalized Riemann–Hilbert problem with irregular singularities. Expo. Math. 24, 235–272 (2006)
B. Dubrovin, M. Mazzocco, Canonical structure and symmetries of the Schlesinger equations. Commun. Math. Phys. 271, 289–373 (2007)
L. Góttsche, Hilbert Schemes of Zero-Dimensional Subschemes of Smooth Varieties, vol. 1572, Lecture Notes in Mathematics (Springer, Berlin, 1994)
V. Heu, Universal isomonodromic deformations of meromorphic rank 2 connections on curves. Ann. Inst. Fourier 60, 515–549 (2010)
Yu. Ilyashenko, S. Yakovenko, Lectures on Analytic Differential Equations, vol. 86, Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2008)
M. Inaba, K. Iwasaki and M.-H. Saito, Dynamics of the sixth Painlevéequation. Théories asymptotiques et équations de Painlevé, 103–167, Sémin. Congr., 14, Soc. Math. France, Paris (2006)
M. Inaba, M.-H. Saito, Moduli of unramified irregular singular parabolic connections on a smooth projective curve. Kyoto J. Math. 53, 433–482 (2013)
M. Inaba, Moduli space of irregular singular parabolic connections of generic ramified type on a smooth projective curve.arXiv:1606.02369
H. Kawamuko, On the Garnier system of half-integer type in two variables. Funkcial. Ekvac. 52, 181–201 (2009)
H. Kimura, The degeneration of the two-dimensional Garnier system and the polynomial Hamiltonian structure. Ann. Mat. Pura Appl. 155, 25–74 (1989)
A. Komyo, Description of generalized isomonodromic deformations of rank two linear differential equations using apparent singularities. arXiv:2003.08045
A. Komyo, M.-H. Saito, Explicit description of jumping phenomena on moduli spaces of parabolic connections and Hilbert schemes of points on surfaces. Kyoto J. Math. 59, 515–552 (2019)
I.M. Krichever, Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. Mosc. Math. J. 2, 717–752 (2002)
F. Loray, M.-H. Saito, C. Simpson, Foliations on the moduli space of rank two connections on the projective line minus four points. Geometric and Differential Galois Theories, 117–170, Sémin. Congr., 27, Soc. Math. France, Paris (2013)
F. Loray, M.-H. Saito, Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line. Int. Math. Res. Not. IMRN 4, 995–1043 (2015)
J. Martinet, J.-P. Ramis, Théorie de Galois différentielle et resommation, Computer Algebra and Differential Equations (Academic Press, London, 1990), pp. 117–214
S. Oblezin, Isomonodromic deformations of sl(2)-Fuchsian systems on the Riemann sphere. Mosc. Math. J. 5, 415–441 (2005)
K. Okamoto, Isomonodromic deformation and Painlevé equations, and the Garnier system. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33, 575–618 (1986)
M. van der Put, M.-H. Saito, Moduli spaces for linear differential equations and the Painlevé equations. Ann. Inst. Fourier (Grenoble) 59, 2611–2667 (2009)
M.-H. Saito, S. Szabó, Apparent singularities and canonical coordinates for moduli of connections. In preparation
S. Szabó, The dimension of the space of Garnier equations with fixed locus of apparent singularities. Acta Sci. Math. (Szeged) 79, 107–128 (2013)
V.S. Varadarajan, Linear meromorphic differential equations: a modern point of view. Bull. Am. Math. Soc. 33, 1–42 (1996)
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We thank CNRS, Université de Rennes 1, Henri Lebesgue Center, ANR-16-CE40-0008 project “Foliage” for financial support and CAPES-COFECUB Ma932/19 project.
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Diarra, K., Loray, F. Normal forms for rank two linear irregular differential equations and moduli spaces. Period Math Hung 84, 303–320 (2022). https://doi.org/10.1007/s10998-021-00408-8
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DOI: https://doi.org/10.1007/s10998-021-00408-8