Holomorphic 𝔰𝔩(2,ℂ)-systems with Fuchsian monodromy (with an appendix by Takuro Mochizuki)

For every integer g≥2 we show the existence of a compact Riemann surface Σ of genus g such that the rank two trivial holomorphic vector bundle 𝒪 Σ ⊕2 admits holomorphic connections with SL(2,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. This also answers a question of Calsamiglia, Deroin, Heu and Loray. The construction carries over to all very stable and compatible real holomorphic structures over the topologically trivial rank two bundle on Σ, and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.