Abstract
The moduli space \({\mathcal {M}}_{g}\), of genus \(g\ge 2\) 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 \(\sigma \). The involution \(\sigma \) maps each point corresponding to a Riemann surface S to its complex conjugate \(\overline{S}\). The fixed point set of \(\sigma \) consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside \(\mathrm {Fix}(\sigma )\) is the locus \({\mathcal {M}}_{g}(\mathbb {R})\), 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 \(\mathrm {Fix}(\sigma )-{\mathcal {M}}_{g}(\mathbb {R})\) 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 \(\mathrm {Fix}(\sigma )\) is connected.
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Partially supported by Project MTM2014-55812-P (Spanish Ministry of Competitiveness), Project of Fondecyt 1150003 and Project Anillo ACT1415 PIA CONICYT.
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Costa, A.F., Hidalgo, R.A. On the connectedness of the set of Riemann surfaces with real moduli. Arch. Math. 110, 305–310 (2018). https://doi.org/10.1007/s00013-017-1132-2
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DOI: https://doi.org/10.1007/s00013-017-1132-2