On the monodromy of holomorphic differential systems

Abstract

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 ${\Sigma }_g$ of genus $g\ge 2$, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian \cite{BDHH} or some cocompact Kleinian subgroup $$\Gamma \subset \text{SL}(2,\mathbb{C})$$ as in \cite{BDHH2}. As a consequence, there exist holomorphic maps from ${\Sigma }_g$ to the quotient space $\text{SL}(2,\mathbb{C})/\Gamma$, where $\Gamma \subset \text{SL}(2,\mathbb{C})$ 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, $\Theta$) 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 $v:\mathcal{O}(\Theta )⟶\mathcal{O}(\varpi )$ of Hopf algebras. We present a description of the transpose of $v$ in terms of the iterated integrals.

To appear in the Int. Jour. Math. volume in honor of Oscar Garcia-Prada