On the generalized Riemann-Hilbert problem with irregular singularities

被引:21
作者
Bolibruch, A. A.
Malek, S.
Mitschi, C.
机构
[1] Univ Strasbourg, Inst Rech Math Avancee, F-67084 Strasbourg, France
[2] VA Steklov Math Inst, Moscow 117966, Russia
[3] Univ Lille 1, UFR Math, F-59655 Villeneuve Dascq, France
[4] CNRS, F-67084 Strasbourg, France
关键词
linear ordinary differential equations; Poincare rank; monodromy; Stokes matrices; Riemann-Hilbert problem; holomorphic vector bundles; connections;
D O I
10.1016/j.exmath.2005.11.003
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We study the generalized Riemann-Hilbert problem, which extends the classical Riemann-Hilbert problem to the case of irregular singularities. The problem is stated in terms of generalized monodromy data which include the monodromy representation, the Stokes matrices and the true Poincare rank at each singular point. We give sufficient conditions for the existence of a linear differential system with such data. These conditions are in particular fulfilled when the monodromy representation is irreducible, as in the classical case. We solve the problem almost completely in dimension two and three. Our results have applications in differential Galois theory. We give sufficient conditions for a given linear algebraic group G to be the differential Galois group over C(z) of a differential system with the smallest possible number of singularities, and with singularities all Fuchsian but one, at which the Poincare rank is minimal. (C) 2005 Elsevier GmbH. All rights reserved.
引用
收藏
页码:235 / 272
页数:38
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