2003/09/18 by Sergey Oblezin, S. Oblezin, Oblezin, S.
Mathematics · Physics and Astronomy · #14E07 #15A54 #32G02 #34B02 #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math-ph #math.MP #math.QA #msc:14E07 #msc:15A54 #msc:32G02 #msc:34B02
paper · pdf · doi:10.48550/arxiv.math-ph/0309048
Revised version, 24 pages
openalex publication_date 2003/09/18 · arxiv created 2005/12/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is devoted to two geometric constructions related to the isomonodromic method. We follow the Drinfeld ideas and develop them in the case of the curve X=ℙ1∖\a1,...,an\. Thus we generalize the results of Arinkin and Lysenko to the case of arbitrary number n of points. First, we construct separated Darboux coordinated in terms of the Hecke correspondences between moduli spaces. In this way we present a geometric interpretation of the Sklyanin formulas. In the second part of the paper, we construct Drinfeld's 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 Painlev´e-VI system.