On the monodromy of holomorphic differential systems

Abstract

First we survey and explain the strategy of some recent results that construct holomorphic $\text{sl}(2,ℂ)$-differential systems over some Riemann surfaces ${\mathrm{\Sigma }}_g$ of genus $g\ge 2$, 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 $$\mathrm{\Gamma }\subset \text{SL}(2,ℂ)$$ as in [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of $\text{SL}(2,ℂ)$, preprint, arXiv:org/abs/2112.03131]. As a consequence, there exist holomorphic maps from ${\mathrm{\Sigma }}_g$ to the quotient space $\text{SL}(2,ℂ)/\mathrm{\Gamma }$, where $\mathrm{\Gamma }\subset \text{SL}(2,ℂ)$ 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 $\text{SL}(2,ℂ)$, 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 $\mathrm{\text{SL}}(2,ℂ)$, 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 $\varpi$ (respectively, $\mathrm{\Theta }$) 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 $v:{𝒪}(\mathrm{\Theta })\to {𝒪}(\varpi )$ of Hopf algebras. We present a description of the transpose of $v$ in terms of the iterated integrals.