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Some special families of rank-2 representations of \(\pi _1\) of compact Riemann surfaces. (English) Zbl 1343.14032

Summary: In this article, we give an explicit way to construct representations of the fundamental group \(\pi _1(X),\) where \(X\) is a hyperbolic curve over \(\mathbb {C}.\) Our motivation is to study a special space in \(M_{\mathrm{DR}} (X,\,\mathrm{SL}_2(\mathbb{C}))\) which is called the space of permissible connections in [G. Faltings, Compos. Math. 48, 223–269 (1983; Zbl 0511.30034)], or indigenous bundles in [R. C. Gunning, Math. Ann. 170, 67–86 (1967; Zbl 0144.33501)]. We get representations by constructing Higgs bundles, and we show that the family we get intersects the space of permissible connections \(\mathbf {PC}\) in a positive dimension. In this way, we actually get a deformation of the canonical representation in \(\mathbf {PC},\) and all these deformations are given by explicit constructed Higgs bundles. We also estimate the dimension of this deformation space.

MSC:

14H60 Vector bundles on curves and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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References:

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