On the monodromy of holomorphic differential systems

Abstract

First we survey and explain the strategy of some recent results that construct holomorphic [Formula: see text]-differential systems over some Riemann surfaces [Formula: see text] of genus [Formula: see text], 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[Formula: see text]-systems of compact Riemann surfaces [with an appendix by Takuro Mochizuki], preprint, arXiv:org/abs/2104.04818] or some cocompact Kleinian subgroup [Formula: see text] as in [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of [Formula: see text], preprint, arXiv:org/abs/2112.03131]. As a consequence, there exist holomorphic maps from [Formula: see text] to the quotient space [Formula: see text], where [Formula: see text] 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 [Formula: see text], 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 [Formula: see text], 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 [Formula: see text] is a Riemann surface, a Torelli-type theorem holds for the affine group scheme over [Formula: see text] 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 [Formula: see text], we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over [Formula: see text]. If [Formula: see text] (respectively, [Formula: see text]) stands for the affine group scheme over [Formula: see text] obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism [Formula: see text] of Hopf algebras. We present a description of the transpose of [Formula: see text] in terms of the iterated integrals.