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Various versions of the Riemann-Hilbert problem for linear differential equations. (English. Russian original) Zbl 1167.34041
Russ. Math. Surv. 63, No. 4, 603-639 (2008); translation from Usp. Mat. Nauk 63, No. 4, 3-42 (2008).
The paper reviews some important results in the Riemann-Hilbert (RH) problem in its various settings including the classical problem of existence of a Fuchsian system and a Fuchsian scalar ODE with given singularities and monodromy, as well as non-classical RH problems on a compact genus \(g\) Riemann surface and a RH problem for a system of ODEs with irregular singularities. The authors treat all the versions of the RH problem using holomorphic vector bundles and meromorphic connections.
In the classical case, the authors show how the Birkhoff-Grothendieck’s theorem results in the Plemelj’s theorem, i.e. why the monodromy representation of the fundamental group of the punctured sphere can always be realized by a system of ODEs that is Fuchsian at all but possibly one point, at which the system is regular. In the case of the RH problem for a scalar Fuchsian equation, the authors formulate a conjecture on a number of additional (apparent) singularities necessary for the RH problem solvability for an arbitrary monodromy representation and give necessary and sufficient conditions of the RH problem solvability in terms of existence of a stable pair of the holomorphic vector bundle of certain splitting type and a meromorphic connection.
It is known that the RH problem on a compact genus \(g\) Riemann surface is solvable in the class of Pfaffian systems with regular singularities if additional (apparent) singularities are allowed. The authors formulate various sufficient conditions for the RH problem solvability and discuss some open questions.
The generalized RH problem with irregular singularities consists in finding of a global system of ODEs with given singular points \(a_1,\dots,a_n\) of given Poincaré ranks \(r_1,\dots,r_n\) and with given monodromy representation of the fundamental group such that, in some neighborhoods of the singular points \(a_i\), this global system is meromorphically equivalent to a local system with a minimal Poincaré rank \(r_i\). The authors formulate an analog of the Plemelj’s theorem, asserting that the generalized monodromy data (i.e. the monodromy representation of the fundamental group and the local systems of minimal Poincaré rank) can be realized by a system of ODEs that has minimal Poincaré ranks at all but possibly one singular point whose Poincaré rank can be estimated using the order of the system, number of singular points and the sum of all assigned Poincaré ranks. Also, the authors provide several sufficient conditions for the solvability of the generalized RH problem and discuss the classical question of existence of a Birkhoff standard form for a system with one Fuchsian and one irregular singular point.

MSC:
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
34A26 Geometric methods in ordinary differential equations
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