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. 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.