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

2021/04/10 by Indranil Biswas, Sorin Dumitrescu, Biswas, Indranil +5
Mathematics · #Mathematical Dynamics and Fractals #Advanced Algebra and Geometry #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2104.04818

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 \mathcal O^{⊕ 2}_Σ admits holomorphic connections with 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 Σ and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.

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