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Mosc. Math. J., 2005, Volume 5, Number 2, Pages 415–441 (Mi mmj202)  

This article is cited in 2 scientific papers (total in 2 papers)

Isomonodromic deformations of $\mathfrak{sl}(2)$ Fuchsian systems on the Riemann sphere

S. V. Oblezin

Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: This paper is devoted to the two geometric constructions provided by the isomonodromic method for Fuchsian systems. We develop the subject in the sense of geometric representation theory following Drinfeld's ideas. Thus we identify the initial data space of the $\mathfrak{sl}(2)$ Schlesinger system with the moduli space of the Frobenius–Hecke (FH-)sheaves originally introduced by Drinfeld. First, we perform the procedure of separation of variables in terms of the Hecke correspondences between moduli spaces. In this way we present a geometric interpretation of the Flashka–McLaughlin, Gaudin and Sklyanin formulas. In the second part of the paper, we construct the Drinfeld compactification of the initial data space and describe the compactifying divisor in terms of certain FH-sheaves. Finally, we give a geometric presentation of the dynamics of the isomonodromic system in terms of deformations of the compactifying divisor and explain the role of apparent singularities for Fuchsian equations. To illustrate the results and methods, we give an example of the simplest isomonodromic system with four marked points known as the Painlevé-VI system.

Key words and phrases: Isomonodromic deformation, separation of variables, the Drinfeld compactification, the Frobenius–Hecke sheaves, the Painlevé-VI equation.

DOI: https://doi.org/10.17323/1609-4514-2005-5-2-415-441

Full text: http://www.ams.org/.../abst5-2-2005.html
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Bibliographic databases:

MSC: 15A54, 32G02, 34B02
Received: February 13, 2004
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Citation: S. V. Oblezin, “Isomonodromic deformations of $\mathfrak{sl}(2)$ Fuchsian systems on the Riemann sphere”, Mosc. Math. J., 5:2 (2005), 415–441

Citation in format AMSBIB
\Bibitem{Obl05}
\by S.~V.~Oblezin
\paper Isomonodromic deformations of $\mathfrak{sl}(2)$ Fuchsian systems on the Riemann sphere
\jour Mosc. Math.~J.
\yr 2005
\vol 5
\issue 2
\pages 415--441
\mathnet{http://mi.mathnet.ru/mmj202}
\crossref{https://doi.org/10.17323/1609-4514-2005-5-2-415-441}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2200759}
\zmath{https://zbmath.org/?q=an:1098.34074}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208595300007}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Levin A.M., Olshanetsky M.A., Zotov A.V., “Painlevé VI, rigid tops and reflection equation”, Comm. Math. Phys., 268:1 (2006), 67–103  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Loray F., Saito M.-H., “Lagrangian Fibrations in Duality on Moduli Spaces of Rank 2 Logarithmic Connections Over the Projective Line”, Int. Math. Res. Notices, 2015, no. 4, 995–1043  crossref  mathscinet  zmath  isi  elib  scopus
  • Moscow Mathematical Journal
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