On the monodromy of holomorphic differential systems (Q6573197)

In recent times, there has been a resurgent interest in the theory of holomorphic differential systems, or connections on trivial bundles, in the special case of a compact ambient space. This interest has been mainly triggered by a strategy of \textit{É. Ghys} [J. Reine Angew. Math. 468, 113--138 (1995; Zbl 0868.32023)] to answer a question raised first by \textit{A. T. Huckleberry} and \textit{J. Winkelmann} [Math. Ann. 295, No. 3, 469--483 (1993; Zbl 0824.32011)]. This question inquires about the existence of holomorphic maps from compact hyperbolic Riemann surfaces to compact complex manifolds of the form \(\mathrm{SL}(2,\mathbb{C})/\Gamma\), with \(\Gamma\) a lattice in \(\mathrm{SL}(2,\mathbb{C})\) (which does not factor to any elliptic curve). This suggests the study of representations of surface groups into \(\mathrm{SL}(2,\mathbb{C})\) giving rise to rank two free (trivial) holomorphic vector bundles over some Riemann surface of genus \(g\geq 2\). So the authors survey and explain how to construct holomorphic \(\mathrm{SL}(2,\mathbb{C})\)-differential systems over some Riemann surfaces \(\Sigma_g\) of genus \(g\geq 2\), satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian or some cocompact Kleinian subgroup \(\Gamma \subset \mathrm{SL}(2,\mathbb{C})\). As a consequence, there exist holomorphic maps from \(\Sigma_g\) to the quotient space \(\mathrm{SL}(2,\mathbb{C})/\Gamma\), where \(\Gamma \subset \mathrm{SL}(2,\mathbb{C})\) is a cocompact lattice, not factoring through any elliptic curve. This answers positively the above question of Ghys, Huckleberry and Winkelmann.\N\NThe authors also 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 étale trivial holomorphic bundles. The authors explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. For a compact Kähler manifold \(M\), they investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over \(M\). If \(\bar{\omega}\) (resp. \(\Theta\)) stands for the affine group scheme over \(\mathbb{C}\) obtained from the category of connections (resp. connections on free (trivial) vector bundles), then the natural inclusion produces a morphism \(v:\mathcal{O}(\Theta )\rightarrow \mathcal{O}(\bar{\omega})\) of Hopf algebras. The authors present a description of the transpose of \(v\) in terms of iterated integrals.