The Connectedness of Some Varieties and the Deligne—Simpson Problem

Abstract

The Deligne--Simpson problem (DSP) (respectively, the weak DSP) is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes Cj � GL(n, �) or cj � gl(n, �) so that there exist irreducible (respectively, 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 the Riemann sphere. For ((p + 1))-tuples of conjugacy classes one of which is with distinct eigenvalues 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 give necessary and sufficient conditions for the positive solvability of the weak DSP.