Geodesic
Stable vector bundles and the Riemann--Hilbert problem on a~Riemann surface
Sbornik. Mathematics, Tome 215 (2024) no. 2, pp. 141-156

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The paper is devoted to holomorphic vector bundles with logarithmic connections on a compact Riemann surface and the applications of the results obtained to the question of solvability of the Riemann–Hilbert problem on a Riemann surface. We give an example of a representation of the fundamental group of a Riemann surface with four punctured points which cannot be realized as the monodromy representation of a logarithmic connection with four singular points on a semistable bundle. For an arbitrary pair of a bundle and a logarithmic connection on it we prove an estimate for the slopes of the associated Harder–Narasimhan filtration quotients. In addition, we present results on the realizability of a representation as a direct summand in the monodromy representation of a logarithmic connection on a semistable bundle of degree zero. Bibliography: 9 titles.
Keywords: Riemann surface, Riemann–Hilbert problem, semistable bundle, logarithmic connection.
Mots-clés : monodromy 
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     author = {I. V. Vyugin and L. A. Dudnikova},
     title = {Stable vector bundles and the {Riemann--Hilbert} problem on {a~Riemann} surface},
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I. V. Vyugin; L. A. Dudnikova. Stable vector bundles and the Riemann--Hilbert problem on a~Riemann surface. Sbornik. Mathematics, Tome 215 (2024) no. 2, pp. 141-156. http://geodesic.mathdoc.fr/item/SM_2024_215_2_a0/