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This is a Quantitative Study study.

October 5, 2025Open Access

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

Key Points

Existence of holomorphic connections on a rank two vector bundle shows the usefulness of Fuchsian monodromy.
A compact Riemann surface of genus g can yield maximal Euler class connections with specific monodromy behaviors.
The study extends insights on holomorphic structures beyond trivial bundles, indicating broader applications in the field.
Answering inquiries posed by Calsamiglia and others, the findings solidify understanding of monodromy representations.

Abstract

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.

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

Key Points

Abstract

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