Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T23:29:25.998Z Has data issue: false hasContentIssue false

Isomonodromic deformation of Fuchsian projective connections on elliptic curves

Published online by Cambridge University Press:  22 January 2016

Shingo Kawai*
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo, 152-8551, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider isomonodromic deformations of second-order Fuchsian differential equations on elliptic curves. The isomonodromic deformations are described as a completely integrable Hamiltonian system.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

References

[1] Biswas, I. and Guruprasad, K., Principal bundles on open surfaces and invariant functions on Lie groups, Internat. J. Math., 4 (1993), 535544.Google Scholar
[2] Goldman, W. M., The symplectic nature of fundamental groups of surfaces, Adv. Math., 54 (1984), 200225.CrossRefGoogle Scholar
[3] Gunning, R. C., Lectures on Riemann surfaces, Math. Notes, No. 2, Princeton Univ. Press, Princeton, NJ, 1966.Google Scholar
[4] Gunning, R. C., Lectures on vector bundles over Riemann surfaces, Math. Notes, No. 6, Princeton Univ. Press, Princeton, NJ, 1967.Google Scholar
[5] Gunning, R. C., Analytic structures on the space of flat vector bundles over a compact Riemann surface, Several complex variables, II (Proceedings of the International Conference, Univ. Maryland, College Park, MD, 1970), pp. 4762, Lecture Notes in Math., Vol. 185, Springer-Verlag, Berlin, 1971.Google Scholar
[6] Iwasaki, K., Moduli and deformation for Fuchsian projective connections on a Riemann surface, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 38 (1991), 431531.Google Scholar
[7] Iwasaki, K., Fuchsian moduli on a Riemann surface — its Poisson structure and Poincaré-Lefschetz duality, Pacific J. Math., 155 (1992), 319340.Google Scholar
[8] Iwasaki, K., Kimura, H., Shimomura, S. and Yoshida, M., From Gauss to Painlevé, Aspects Math., E16, Vieweg, Braunschweig, 1991.Google Scholar
[9] Kawai, S., Deformation of complex structures on a torus and monodromy preserving deformation, Thesis (1995), University of Tokyo.Google Scholar
[10] Kawai, S., The symplectic nature of the space of projective connections on Riemann surfaces, Math. Ann., 305 (1996), 161182.CrossRefGoogle Scholar
[11] Kawai, S., Isomonodromic deformation of Fuchsian-type projective connections on elliptic curves, Analysis of discrete groups, II (Kyoto, 1996), Sūrikaisekikenkyūsho Kōkyūroku, 1022 (1997), 5357.Google Scholar
[12] Kawai, S., A remark on Manin’s description of the sixth Painlevé equation, in preparation.Google Scholar
[13] Korotkin, D. A. and Samtleben, H., On the quantization of isomonodromic deformations on the torus, Internat. J. Modern Phys., A 12 (1997), 20132029, hep-th/9511087.Google Scholar
[14] Levin, A. M. and Olshanetsky, M. A., Hierarchies of isomonodromic deformations and Hitchin systems, Moscow Seminar in Mathematical Physics, pp. 223262, Amer. Math. Soc. Transl. Ser. 2, Vol. 191, Amer. Math. Soc., Providence, RI, 1999.Google Scholar
[15] Manin, Yu. I., Sixth Painlevé equation, universal elliptic curve, and mirror of P2 , Geometry of differential equations, pp. 131151, Amer. Math. Soc. Transl. Ser. 2, Vol. 186, Amer. Math. Soc., Providence, RI, 1998, alg-geom/9605010.Google Scholar
[16] Okamoto, K., On Fuchs’s problem on a torus, I, Funkcial. Ekvac, 14 (1971), 137152.Google Scholar
[17] Okamoto, K., The Hamiltonian structure derived from the holonomic deformation of the linear ordinary differential equations on an elliptic curve, Sci. Papers College Arts Sci. Univ. Tokyo, 37 (1987), 111.Google Scholar
[18] Okamoto, K., On the holonomic deformation of linear ordinary differential equations on an elliptic curve, Kyushu J. Math., 49 (1995), 281308.CrossRefGoogle Scholar
[19] Takasaki, K., Gaudin model, KZ equation, and isomonodromic problem on torus, Lett. Math. Phys., 44 (1998), 143156, hep-th/9711058.CrossRefGoogle Scholar
[20] Takasaki, K., Elliptic Calogero-Moser systems and isomonodromic deformations, J. Math. Phys., 40 (1999), 57875821, math.QA/9905101.Google Scholar