Regular Linear System Fuchsian System Monodromy Group
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 27, No 2, (2001), 143p-158p
We consider the variety of (p + 1)-tuples of matrices Aj (resp.
Mj ) from given conjugacy classes cj ⊂ gl(n, C) (resp. Cj ⊂ GL(n, C))
such that A1 + . . . + A[p+1] = 0 (resp. M1 . . . M[p+1] = I). This variety is
connected with the weak Deligne-Simpson problem: give necessary and sufficient
conditions on the choice of the conjugacy classes cj ⊂ gl(n, C) (resp.
Cj ⊂ GL(n, C)) so that there exist (p + 1)-tuples with trivial centralizers of
matrices Aj ∈ cj (resp. Mj ∈ Cj ) whose sum equals 0 (resp. whose product
equals I). The matrices Aj (resp. Mj ) are interpreted as matrices-residua
of Fuchsian linear systems (resp. as monodromy operators of regular linear
systems) on Riemann’s sphere. We consider examples of such varieties of
dimension higher than the expected one due to the presence of (p + 1)-tuples
with non-trivial centralizers; in one of the examples the difference between
the two dimensions is O(n).
Research partially supported by INTAS grant 97-1644