The connectedness of some varieties and the Deligne-Simpson problem

The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this: give necessary and sufficient conditions for the choice of the conjugacy classes Cj ⊂ GL(n,C) or cj ⊂ gl(n,C) so that there exist irreducible (resp. with trivial centralizer) (p + 1)-tuples of matrices Mj ∈ Cj or Aj ∈ cj satisfying the equality M1 . . .Mp+1 = I or A1+ . . .+Ap+1 = 0. The matrices Mj and Aj 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 {(M1, . . . ,Mp+1)|Mj ∈ Cj ,M1 . . .Mp+1 = I} or {(A1, . . . , Ap+1)|Aj ∈ cj , A1 + . . .+Ap+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.