Real hyperelliptic surfaces and the orbifold fundamental group

In this paper we finish the topological classification of real algebraic surfaces of Kodaira dimension zero and we make a step towards the Enriques classification of real algebraic surfaces, by describing in detail the structure of the moduli space of real hyperelliptic surfaces. Moreover, we point out the relevance in real geometry of the notion of the orbifold fundamental group of a real variety, and we discuss related questions on real varieties $(X, \sigma)$ whose underlying complex manifold $X$ is a $K (\pi, 1)$. Our first result is that if $(S, \sigma)$ is a real hyperelliptic surface, then the differentiable type of the pair $(S, \sigma)$ is completely determined by the orbifold fundamental group exact sequence. This result allows us to determine all the possible topological types of $(S, \sigma)$, and to prove that they are exactly 78. It follows also as a corollary that there are exactly eleven cases for the topological type of the real part of S. Finally, we show that once we fix the topological type of $(S, \sigma)$ corresponding to a real hyperelliptic surface, the corresponding moduli space is irreducible (and connected). We also give, through a series of tables, explicit analytic representations of the 78 components of the moduli space.