Gauss–Manin Systems of Polynomials of Two Variables can be Made Fuchsian

Abstract

We prove, modulo a conjecture due to A. A. Bolibrukh, that every monodromy group in which the operators of local monodromy in their Jordan normal forms have Jordan blocks of size only $\leq 2$ can be realized by a Fuchsian system of linear differential equations on Riemann's sphere without additional apparent singularities. This implies that the Gauss-Manin system of a polynomial of two variables can always be made Fuchsian if a suitable basis in the cohomologies is chosen.