Some Special Families of Rank-2 Representations of $$\pi _1$$ π 1 of Compact Riemann Surfaces

Abstract

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 Faltings (Compos Math 48(2):223–269, 1983), or indigenous bundles in Gunning (Math Ann 170:67–86, 1967). 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.