On the monodromy of holomorphic differential systems
First we survey and explain the strategy of some recent results that construct holomorphic $\text{sl}(2, \mathbb C)$-differential systems over some Riemann surfaces $Σ_g$ of genus $g\geq 2$, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian \cite{BDHH} or some cocompact Kleinian subgroup $$Γ\subset \text{SL}(2, \mathbb C)$$ as in \cite{BDHH2}. As a consequence, there exist holomorphic maps from $Σ_g$ to the quotient space $\text{SL}(2, \mathbb C)/ Γ$, where $Γ\subset \text{SL}(2, \mathbb C)$ is a cocompact lattice, that do not factor through any elliptic curve \cite{BDHH2}. This answers positively a question of Ghys in \cite{Gh}; the question was also raised by Huckleberry and Winkelmann in \cite{HW}. Then we prove that when $M$ is a Riemann surface, a Torelli type theorem holds for the affine group scheme over $\mathbb C$ obtained from the category of holomorphic connections on {\it étale 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 Kähler manifold $M$, we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over $M$. If $\varpi$ (respectively, $Θ$) stands for the affine group scheme over $\mathbb C$ obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism $v:{\mathcal O}(Θ)\longrightarrow {\mathcal O}(\varpi)$ of Hopf algebras. We present a description of the transpose of $v$ in terms of the iterated integrals.