Abstract
We explain the strategy of some recent results that construct holomorphic sl(2, C)-differential systems over some Riemann surfaces Σ g of genus g ≥ 2, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian [BDHH1] or some cocompact Kleinian subgroup Γ ⊂ SL(2, C) [BDHH2]. As a consequence, there exist holomorphic maps from Σ g to the quotient space SL(2, C)/Γ, where Γ ⊂ SL(2, C) is a cocompact lattice, that do not factor through any elliptic curve [BDHH2]. This answers positively a question of Ghys in [Gh]; the question was also raised by Huckleberry and Winkelmann in [HW]. When M is a Riemann surface, a Torelli type theorem holds for the affine group scheme over C obtained from the category of holomorphic connections on étale trivial holomorphic bundles. We explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. For a compact Kähler manifold M , we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over M. If (respectively, Θ) stands for the affine group scheme over C obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism v : O(Θ) −→ O() of Hopf algebras. We present a description of the transpose of v in terms of the iterated integrals.