The Deligne–Simpson Problem and the Connectedness of some Varieties

Abstract

The Deligne---Simpson problem is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes $C_{j} \subset {\text{GL}}{\left( {n,\mathbb{C}} \right)}$ or $c_{j} \subset {\text{gl}}{\left( {n,\mathbb{C}} \right)}$ so that there exist irreducible (p+1)-tuples of matrices M j ? C j or A j ? 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-residues of Fuchsian ones on the Riemann sphere. We prove that in the so-called simple case the subset $\mathcal{W}^{\prime }$ or $\mathcal{V}^{\prime }$ of the variety $\mathcal{W}: = {\left\{ {{\left( {M_{1} , \ldots ,M_{{p + 1}} } \right)}\left| {M_{j} \in C_{j} ,M_{1} \cdots M_{{p + 1}} = I} \right.} \right\}}$ or $\mathcal{V}: = {\left\{ {{\left( {A_{1} , \ldots ,A_{{p + 1}} } \right)}\left| {A_{j} \in c_{j} ,A_{1} + } \right. \cdots + A_{{p + 1}} = 0} \right\}}$ consisting of all irreducible (p+1)-tuples (if nonempty) is connected. "Simple" means that the greatest common divisor of all quantities of Jordan blocks of a given size, of a given matrix M j or A j , and with a given eigenvalue is 1.