On the connectedness of the set of Riemann surfaces with real moduli

2017/10/12 by Costa, Antonio F., Hidalgo, Ruben A.
#30F10 #30F60 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.04627

Abstract

The moduli space M_g, of genus g≥2 closed Riemann surfaces, is a complex orbifold of dimension 3(g-1) which carries a natural real structure i.e. it admits an anti-holomorphic involution σ. The involution σ maps each point corresponding to a Riemann surface S to its complex conjugate S. The fixed point set of σ consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside Fix(σ) is the locus M_g(ℝ), the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Seppälä and R. Silhol. The complement Fix(σ)-M_g(ℝ) consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that Fix(σ) is connected.

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