Holomorphic $\mathfrak{sl}(2,\mathbb C)$-systems with Fuchsian monodromy (with an appendix by Takuro Mochizuki)

TL;DR

This paper proves the existence of special holomorphic connections with Fuchsian monodromy on certain rank two bundles over Riemann surfaces of genus at least 2, extending known uniformization results.

Contribution

It constructs holomorphic connections with Fuchsian monodromy on trivial rank two bundles over Riemann surfaces of genus ≥ 2, including stable and compatible real structures.

Findings

Existence of holomorphic connections with SL(2,R) monodromy on trivial bundles.

Construction applies to all very stable and compatible real structures.

Connections realize Fuchsian uniformization for genus g surfaces.

Abstract

For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $\Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_{\Sigma}$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $\Sigma$ and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.