The connectedness of some varieties and the Deligne-Simpson problem

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 $C_j\subset GL(n,{\bf C})$ or $c_j\subset gl(n,{\bf C})$ 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.