Abstract
The solvability of the Riemann-Hilbert problem for representations χ = χ 1 ⊕ χ 2 having the form of a direct sum is considered. It is proved that any representation χ 1 can be realized as a direct summand in the monodromy representation χ of a Fuchsian system. Other results are also obtained, which suggest a simple method for constructing counterexamples to the Riemann-Hilbert problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. A. Bolibrukh, Fuchsian Differential Equations and Holomorphic Bundles, in Modern Lecture Courses (MTsNMO, Moscow, 2000) [in Russian].
A. A. Bolibrukh, Hilbert’s Twenty-First Problem for Linear Fuchsian Systems, inTrudy Mat. Inst. Steklov (Nauka, Moscow, 1994), Vol. 206 [Proc. Steklov Inst.Math. 206 (5), (1995)].
I. V. V’yugin, “An indecomposable Fuchsian system with a decomposable monodromy representation,” Mat. Zametki 80(4), 501–508 (2006) [Math. Notes 80 (3–4), 478–484 (2006)].
A. A. Bolibrukh, “The Riemann-Hilbert problem on a compact Riemann surface,” in Trudy Mat. Inst. Steklov, Vol. 238: Monodromy in Problems of Algebraic Geometry and Differential Equations (Nauka, Moscow, 2002), pp. 55–69 [Proc. Steklov Inst.Math. 238 (3), 47–60 (2002)].
H. Esnault and C. Hertling, “Semistable bundles on curves and reducible representations of the fundamental group,” Internat. J. Math. 12(7), 847–855 (2001).
Author information
Authors and Affiliations
Additional information
Original Russian Text © I. V. V’yugin, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 6, pp. 817–825.
Rights and permissions
About this article
Cite this article
V’yugin, I.V. Fuchsian systems with completely reducible monodromy. Math Notes 85, 780–786 (2009). https://doi.org/10.1134/S0001434609050204
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434609050204