Some Special Families of Rank-2 Representations of $\pi _1$ 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.