Isomonodromic deformations of \({\mathfrak {sl}}(2)\) Fuchsian systems on the Riemann sphere. (English) Zbl 1098.34074

Oblezin, Sergey

Mosc. Math. J. 5, No. 2, 415-441 (2005).

The author discusses a geometry related to the isomonodromic deformations of (nonresonant) rank 2 Fuchsian systems with \(n\) singular points \(a_i\), \(i=1,\dots,n\). The fundamental matrix to the latter is described in terms of horizontal sections of a certain rank-2 bundle with respect to a logarithmic \(sl(2)\)-connection on the Riemann sphere. In this way, the author identifies the phase space of a Fuchsian system, i.e., the initial data space, with the phase space of the isomonodromic deformations of dimension \(2(n-3)\), or Frobenius-Hecke sheaves introduced by Drinfeld. The conventional procedure of separation of variables yields the system of the “étale” coordinates \(\{x_i,p_i\}\), \(i=1,\dots,n-3\), on the phase space \({\mathcal M}_n\). The author discusses the behavior of the system on the divisors \(\{x_i=a_j\}\) and shows which obstacles arise on diagonals \(\{x_i=x_j\}\). At the remaining part of the paper, the author describes the Drinfeld compactification of the initial data space in terms of the introduced variables. In particular, he discusses the structure of the compactifying divisor \({\mathbf \Theta}_n\), describes the dynamical variables \(\{x_i,p_i\}\) as the parameters of the Hecke correspondence between \({\mathbf \Theta}_n\) and \({\mathcal M}_n\) and shows that the parameters \(\{x_i\}\) turn into apparent singularities of the relevant connection. In conclusion, the author illustrates the presentation by the geometry of the Painlevé-VI system.

Reviewer: Andrei A. Kapaev (St. Petersburg)

MSC:

34M55	Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
32G34	Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)

Keywords:

isomonodromic deformation; separation of variables; the Drinfeld compactification; the Frobenius-Hecke sheaves; the Painlevé-VI equation