Ramified covers and tame isomonodromic solutions on curves

Diarra, Karamoko; Loray, Frank

doi:10.1090/mosc/247

Ramified covers and tame isomonodromic solutions on curves
HTML articles powered by AMS MathViewer

by Karamoko Diarra and Frank Loray

Trans. Moscow Math. Soc. 2015, 219-236

DOI: https://doi.org/10.1090/mosc/247

Published electronically: November 18, 2015

PDF

Abstract:

In this paper, we investigate the possibility of constructing isomonodromic deformations by ramified covers. We give new examples and prove a classification result.

References

F. V. Andreev and A. V. Kitaev, Transformations $RS^2_4(3)$ of the ranks $\leq 4$ and algebraic solutions of the sixth Painlevé equation, Comm. Math. Phys. 228 (2002), no. 1, 151–176. MR 1911252, DOI 10.1007/s002200200653
Philip Boalch, Towards a non-linear Schwarz’s list, The many facets of geometry, Oxford Univ. Press, Oxford, 2010, pp. 210–236. MR 2681697, DOI 10.1093/acprof:oso/9780199534920.003.0011
Bolibrukh A. A., The Riemann–Hilbert problem, Russian Math. Surveys. Vol. 45, No. 2 (1990), 1–58.
Serge Cantat and Frank Loray, Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 7, 2927–2978 (English, with English and French summaries). MR 2649343
Karamoko Diarra, Construction et classification de certaines solutions algébriques des systèmes de Garnier, Bull. Braz. Math. Soc. (N.S.) 44 (2013), no. 1, 129–154 (French, with English and French summaries). MR 3077637, DOI 10.1007/s00574-013-0006-x
Karamoko Diarra, Solutions algébriques partielles des équations isomonodromiques sur les courbes de genre 2, Ann. Fac. Sci. Toulouse Math. (6) 24 (2015), no. 1, 39–54 (French, with English and French summaries). MR 3325950, DOI 10.5802/afst.1441
Charles F. Doran, Algebraic and geometric isomonodromic deformations, J. Differential Geom. 59 (2001), no. 1, 33–85. MR 1909248
Boris Dubrovin and Marta Mazzocco, On the reductions and classical solutions of the Schlesinger equations, Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys., vol. 9, Eur. Math. Soc., Zürich, 2007, pp. 157–187. MR 2322330
Viktoria Heu, Universal isomonodromic deformations of meromorphic rank 2 connections on curves, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 2, 515–549 (English, with English and French summaries). MR 2667785
Heu V., Loray F., Flat rank 2 vector bundles on genus 2 curves. arXiv:1401.2449 [math.AG].
N. J. Hitchin, Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differential Geom. 42 (1995), no. 1, 30–112. MR 1350695
A. V. Kitaev, Quadratic transformations for the sixth Painlevé equation, Lett. Math. Phys. 21 (1991), no. 2, 105–111. MR 1093520, DOI 10.1007/BF00401643
I. Krichever, Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations, Mosc. Math. J. 2 (2002), no. 4, 717–752, 806 (English, with English and Russian summaries). Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1986088, DOI 10.17323/1609-4514-2002-2-4-717-752
Oleg Lisovyy and Yuriy Tykhyy, Algebraic solutions of the sixth Painlevé equation, J. Geom. Phys. 85 (2014), 124–163. MR 3253555, DOI 10.1016/j.geomphys.2014.05.010
F. Loray, M. van der Put, and F. Ulmer, The Lamé family of connections on the projective line, Ann. Fac. Sci. Toulouse Math. (6) 17 (2008), no. 2, 371–409 (English, with English and French summaries). MR 2487859
Frank Loray, Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections, Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, pp. 129–136. MR 2310025
Loray F., Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symmetry. arXiv:1410.4976 [math.AG] (2014).
Francois-Xavier Machu, Monodromy of a class of logarithmic connections on an elliptic curve, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 082, 31. MR 2366940, DOI 10.3842/SIGMA.2007.082
Yu. I. Manin, Sixth Painlevé equation, universal elliptic curve, and mirror of $\mathbf P^2$, Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, vol. 186, Amer. Math. Soc., Providence, RI, 1998, pp. 131–151. MR 1732409, DOI 10.1090/trans2/186/04
Marta Mazzocco, Picard and Chazy solutions to the Painlevé VI equation, Math. Ann. 321 (2001), no. 1, 157–195. MR 1857373, DOI 10.1007/PL00004500
Marta Mazzocco, The geometry of the classical solutions of the Garnier systems, Int. Math. Res. Not. 12 (2002), 613–646. MR 1892536, DOI 10.1155/S1073792802106118
Marta Mazzocco and Raimundas Vidunas, Cubic and quartic transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence, Stud. Appl. Math. 130 (2013), no. 1, 17–48. MR 3010489, DOI 10.1111/j.1467-9590.2012.00562.x
Kazuo Okamoto, Isomonodromic deformation and Painlevé equations, and the Garnier system, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986), no. 3, 575–618. MR 866050
Kazuo Okamoto and Hironobu Kimura, On particular solutions of the Garnier systems and the hypergeometric functions of several variables, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 61–80. MR 830631, DOI 10.1093/qmath/37.1.61
Carlos T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 5–79 (1995). MR 1320603
Teruhisa Tsuda, Kazuo Okamoto, and Hidetaka Sakai, Folding transformations of the Painlevé equations, Math. Ann. 331 (2005), no. 4, 713–738. MR 2148794, DOI 10.1007/s00208-004-0600-8
Humihiko Watanabe, Birational canonical transformations and classical solutions of the sixth Painlevé equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), no. 3-4, 379–425 (1999). MR 1678014