RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mosc. Math. J.: Year: Volume: Issue: Page: Find

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

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
References: PDF file   HTML file

Bibliographic databases:

MSC: 15A54, 32G02, 34B02
Language:

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} 

• http://mi.mathnet.ru/eng/mmj202
• http://mi.mathnet.ru/eng/mmj/v5/i2/p415

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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