Gauss-Manin systems of polynomials of two variables can be made Fuchsian. (English) Zbl 0979.34066

Kostov, Vladimir P.

Mladenov, I. M. (ed.) et al., Proceedings of the international conference on geometry, integrability and quantization, Varna, Bulgaria, September 1-10, 1999. Sofia: Coral Press Scientific Publishing. 105-126 (2000).

Consider a system of linear equations \(dx/dt= A(t)x\) (\(A\) is a matrix meromorphic in the projective line), which is regular-singular at each pole \(a_j\) of \(A\) and at infinity. Such a system is called Fuchsian if \(A\) is of the form \(\sum A_j/(t- a_j)\) with constant \(A_j\). Here, the following Riemann-Hilbert-type theorem is proved modulo a conjecture due to A. A. Bolibrukh: If at every \(a_j\) local monodromy in its Jordan form has only Jordan blocks of size \(\leq 2\), then the monodromy group is realizable by a Fuchsian system for any prescribed set of poles.
The theorem would lead to the following: Any Gauss-Manin system of a polynomial of two variables can be made Fuchsian if one chooses a suitable basis in the cohomologies.
For the entire collection see [Zbl 0940.00039].

Reviewer: Masaaki Yoshida (Fukuoka)

MSC:

34M15	Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of 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
34M25	Formal solutions and transform techniques for ordinary differential equations in the complex domain

Keywords:

local monodromy; monodromy group; Gauss-Manin system; cohomologies