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 Σ g of genus g ≥ 2 , satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, Fuchsian sl (2 , ℂ) -systems of compact Riemann surfaces [with an appendix by Takuro Mochizuki], preprint, arXiv:org/abs/2104.04818] or some cocompact Kleinian subgroup Γ ⊂ SL (2 , ℂ) as in [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. As a consequence, there exist holomorphic maps from Σ g 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 [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. This answers positively a question of Ghys in [E. Ghys, Déformations des structures complexes sur les espaces homogènes de SL (2 , ℂ) , J. Reine Angew. Math. 468 (1995) 113–138]; the question was also raised by Huckleberry and Winkelmann in [A. H. Huckleberry and J. Winkelmann, Subvarieties of parallelizable manifolds, Math. Ann. 295 (1993) 469–483]. 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 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 ϖ (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 : (Θ) → (ϖ) of Hopf algebras. We present a description of the transpose of v in terms of the iterated integrals. [ABSTRACT FROM AUTHOR]