Journal of Dynamical and Control Systems

, Volume 11, Issue 1, pp 125–155 | Cite as

The Connectedness of Some Varieties and the Deligne—Simpson Problem

Article

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, ℂ) or cj ⊂ gl(n, ℂ) so that there exist irreducible (respectively, with trivial centralizer) (p + 1)-tuples of matrices MjCj or Ajcj 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) | MjCj, M1 ... Mp+1 = I} or {(A1, ..., Ap+1) | Ajcj, 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.

Key words and phrases:

Generic eigenvalues monodromy operator (weak) Deligne—Simpson problem 

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Copyright information

© Springer Science+Business Media, Inc 2005

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de Nice, Sophia AntipolisNice, Cedex 2France

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