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

Abstract

For every integer $g\ge 2$ we show the existence of a compact Riemann surface $\Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal{O}}_{\Sigma }^{\oplus 2}$ 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.

38 pages, 2 figures