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

TL;DR

This paper proves that the set of all Riemann surfaces with real moduli, including real and pseudo-real surfaces, forms a connected set within the moduli space, clarifying the topological structure of these surfaces.

Contribution

The paper presents a simple argument demonstrating the connectedness of the entire fixed point set of the real structure in the moduli space, encompassing both real and pseudo-real Riemann surfaces.

Findings

The fixed point set of the real structure is connected.

Real Riemann surfaces form a connected subset of the moduli space.

Pseudo-real Riemann surfaces are included in the connected set.

Abstract

The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ 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. Sepp\"{a}l\"{a} 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.