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

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. Seppala, 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.