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



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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  1. 1.
    1. V. I. Arnold and V. I. Ilyashenko, Ordinary differential equations. In Dynamical Systems I, Encyclopaedia of Mathematical Sciences 1, Springer-Verlag (1988).Google Scholar
  2. 2.
    2. A. A. Bolibrukh, Hilbert’s twenty-first problem for linear Fuchsian systems. Proc. Steklov Inst. Math. 206 (1995), No. 5.Google Scholar
  3. 3.
    3. A. A. Bolibrukh, The Riemann—Hilbert problem. Russ. Math. Surv. 45 (1990), No. 2, 1–49.Google Scholar
  4. 4.
    4. W. Crawley-Boevey, On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero. Preprint arXiv:math.LA/0103101 (2001); to appear in Duke Math. J. 118 (2003).Google Scholar
  5. 5.
    5. F. R. Gantmacher, The theory of matrices. Vol. 1, 2. Chelsea Publ., New York (1959).Google Scholar
  6. 6.
    6. W. Hesselink, Singularities in the nilpotent scheme of a classical group. Trans. Amer. Math. Soc. 222 (1976), 1–32.Google Scholar
  7. 7.
    7. N. M. Katz, Rigid local systems. Ann. Math. Stud. Ser. 139 (1995).Google Scholar
  8. 8.
    8. V. P. Kostov, The Deligne—Simpson problem. C. R. Acad. Sci. Paris 329 (1999), 657–662.Google Scholar
  9. 9.
    9. V. P. Kostov, On the Deligne—Simpson problem. Electronic preprint math.AG/0011013; to appear in Proc. Steklov Math. Inst. 238 (2002).Google Scholar
  10. 10.
    10. V. P. Kostov, On some aspects of the Deligne—Simpson problem. J. Dynam. Control Systems 9 (2003), No. 3, 393–436; Preprint math.AG/0005016.Google Scholar
  11. 11.
    11. V. P. Kostov, Monodromy groups of regular systems on Riemann’s sphere. Preprint No. 401, Univ. de Nice (1994).Google Scholar
  12. 12.
    12. V. P. Kostov, Some examples of rigid representations. Serdica Math. J. 26 (2000), 253–276.Google Scholar
  13. 13.
    13. V. P. Kostov, Quantum states of monodromy group. J. Dynam. Control Systems 5 (1999), No. 1, 51–100.Google Scholar
  14. 14.
    14. V. P. Kostov, Some examples related to the Deligne—Simpson problem. Preprint math.AG/0011015.Google Scholar
  15. 15.
    15. V. P. Kostov, Examples illustrating some aspects of the Deligne—Simpson problem. Serdica Math. J. 27 (2001), No. 2, 143–158; Preprint math.AG0101141.Google Scholar
  16. 16.
    16. H. Kraft, Parametrisierung von Konjugationsklassen in sl. Math. Ann. 234 (1978), 209–220.Google Scholar
  17. 17.
    17. C. T. Simpson, Products of matrices. In: Differential Geometry, Global Analysis, and Topology, Canad. Math. Soc. Conference Proceedings 12, AMS, Providence (1992), 157–185.Google Scholar

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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