The connectedness of some varieties and the Deligne-Simpson problem
  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. 
Vladimir Petrov Kostov