The connectedness of some varieties and the Deligne-Simpson problem

Vladimir Petrov Kostov

paper

The connectedness of some varieties and the Deligne-Simpson problem

arXiv:math/0206087

Abstract

The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_{j} \subset G L (n, C)$ or $c_{j} \subset g l (n, C)$ so that there exist irreducible (resp. with trivial centralizer) $(p + 1)$ -tuples of matrices $M_{j} \in C_{j}$ or $A_{j} \in c_{j}$ satisfying the equality $M_{1} ... M_{p + 1} = I$ or $A_{1} + ... + A_{p + 1} = 0$ }. The matrices $M_{j}$ and $A_{j}$ are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on Riemann's sphere. For $(p + 1)$ -tuples of conjugacy classes one of which is with distinct eigenvalues 1) we prove that the variety ${(M_{1}, ..., M_{p + 1}) ∣ M_{j} \in C_{j}, M_{1} ... M_{p + 1} = I}$ or ${(A_{1}, ..., A_{p + 1}) ∣ A_{j} \in c_{j}, A_{1} + ... + A_{p + 1} = 0}$ is connected if the DSP is positively solved for the given conjugacy classes and 2) we give necessary and sufficient conditions for the positive solvability of the weak DSP.