On the monodromy of holomorphic differential systems

First we survey and explain the strategy of some recent results that construct holomorphic sl(2, C)-differential systems over some Riemann surfaces Σg of genus g≥ 2, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian {BDHH} or some cocompact Kleinian subgroup Γ ⊂ SL(2, C) as in {BDHH2}. As a consequence, there exist holomorphic maps from Σg 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}. Then we prove that 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 { \'etale trivial} holomorphic bundles. After that, we explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. Finally, for a compact K\"ahler manifold M, we investigate the neutral Tannakian category given by the holomorphic connections on \'etale 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.