Sergei Yakovenko's blog: on Math and Teaching

Tuesday, December 30, 2014

Lecture 12+ (Mon, Dec 22, 2014)

Riemann-Hilbert problem

The Riemann-Hilbert problem consists in “constructing a Fuchsian system with a prescribed monodromy”.

More precisely, let M_1,M_2,\dots,M_d be nondegenerate matrices such that their product is an identical matrix, and a_0,a_1,\dots, a_d\in\mathbb C are distinct points, such that the segments [a_0,a_k]\subset\mathbb C,\ k=1,\dots,d are all disjoint except for the point a_0 itself.

The problem is to construct a linear system of equations

\displaystyle \dot X=A(t)X,\quad A(t)=\sum_{k=1}^d \frac{A_k}{t-a_k},\quad \sum_{k=1}^d A_k=0,

such that the monodromy operator along the path “\gamma_k=segment [a_0,a_k]+ small loop around a_k+segment [a_k,a_0]” is equal to M_k.

The modern strategy of solving this problem is surgery. One can easily construct a local solution, a differential system on a neighborhood U_k of the segment [a_0,a_k], which has the specified monodromy. The phase space of this system is the cylinder U_k\times\mathbb C^n, and without loss of generality one can assume that together the neighborhoods U_k cover the whole Riemann sphere \mathbb CP^1=\mathbb C\cup\{\infty\}. Patching together these local solutions, one can construct a linear system with the specified monodromy, but it will be defined not on \mathbb C P^1\times\mathbb C^n, as required, but on a more general object, holomorphic vector bundle over \mathbb C P^1.

Description of different vector bundles is of an independent interest and is well known. It turns out (Birkhoff), that each holomorphic vector bundle in dimension n is completely determined by a(n unordered) tuple of integer numbers d_1,\dots,d_n\in\mathbb Z, and the bundle is trivial if and only if d_1=\cdots=d_n=0.

However, the strategy of solving the Riemann-Hilbert problem by construction of the bundle and determining its holomorphic type is complicated by two facts:

  1. Determination of the holomorphic type of a bundle is a transcendental problem;
  2. The local realization of the monodromy is by no means unique: in the non-resonant case one can realize any matrix M_k by an Euler system with the eigenvalues which can be arbitrarily shifted by integers; in the resonant case one should add to this freedom also non-Euler systems. This freedom can change the holomorphic type of the vector bundle in a very broad range.

It turns out that the fundamental role in solvability of the Riemann-Hilbert problem plays the (ir)reducibility of the linear group generated by the matrices M_1,\dots,M_k.

Theorem (Bolibruch, Kostov). If the group is irredicible, i.e., there is no invariant subspace in \mathbb C^n common for all operators M_k, then one can choose the local realizations in such a way that the resulting bundle is trivial and thus yields solution to the Riemann-Hilbert problem. 

The proof is achieved as follows: one constructs a possibly nontrivial bundle realizing the given monodromy, and then this bundle is brutally trivialized by a transformation that is only meromorphic at one of the singularities. The result will be a system with all but one singularities being Fuchsian, and the problem reduces to bringing to the Fuchsian form the last point (assumed to be at infinity) by transformations of the form X\mapsto P(t)X with P being a matrix polynomial with a constant nonzero determinant.  The group of such transformations is considerably more subtle, but ultimately the freedom in construction of the initial bundle can be used to guarantee that the last point is also “Fuchsianizible”.

All the way around, if the monodromy group is reducible, then there is an obstruction of the torsion type exists for trivializing the bundle. This obstruction was first discovered by A. Bolibruch, and its description can be found in the textbook by Yu. Ilyashenko and SY (sections 16G and 18).

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