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.
Categories: math.CV