Studies on monodromy preserving deformation of linear differential equations on elliptic curves

Abstract

We study a monodromy preserving deformation ͑MPD͒ of linear differential equations on elliptic curves. As the first of our results, we describe asymptotic behaviors of solutions to the MPD system when the elliptic curve degenerates to a rational curve. As the second, we find explicit solutions for special values of parameters where the MPD system is linearizable. Our solutions are written in terms of integrals of theta functions. We also show that they converge to the hypergeometric functions applying the above asymptotic formula when the elliptic curve degenerates to a rational curve.

The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces

“…In the case C 2 = · · · = C k = 0 the derivation above just reproduce the derivation of the Schlesinger equations for the deformations of the form (2) (see [1]). In particular in notations a 0 = 0, B 0 = C 1 = − n i=1 B i we obtain an ordinary Schlesinger system (9). The proposition is proved.…”

“…It is also natural to allow to change a module of a complex structure. Such deformations were considered by different authors in [3]- [9] and also in many other papers.…”

Система Шлезингера И Изомонодромные Деформации Расслоений Со Связностями На Римановых Поверхностях

“…Также естественно допускать деформацию модуля комплексной структуры. Подобные деформации рассматривались различными авторами в работах [3]- [9] и во многих других работах.…”

“…монографию [1]). Именно поэтому в обозначениях a 0 = 0, B 0 = C 1 = − n i=1 B i получается обычная система Шлезингера (9). Предложение доказано.…”

Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral

“…According to Mano [14], [15], the integral (1.1) appears as a particular solution of a system of partial differential equations of the integrability condition of monodromy-preserving deformation of a Fuchsian differential equation with n singularities t 1 , . .…”