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 g of genus g 2, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian BDHH or some cocompact Kleinian subgroup SL (2, C) as in BDHH2. As a consequence, there exist holomorphic maps from g to the quotient space SL (2, C) /, where 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, ) 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 () O () of Hopf algebras. We present a description of the transpose of v in terms of the iterated integrals.