, Volume 11, Issue 1, pp 125-155
Date: 25 Jan 2005

The Connectedness of Some Varieties and the Deligne—Simpson Problem

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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 C j ⊂ GL(n, ℂ) or c j ⊂ gl(n, ℂ) so that there exist irreducible (respectively, with trivial centralizer) (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-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 {(M 1, ..., M p+1) | M j C j , M 1 ... M p+1 = I} or {(A 1, ..., A p+1) | A j c j , A 1 + ... + A p+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.

2000 Mathematics Subject Classification. 15A30, 15A24, 20G05.