Real Kodaira surfaces

In this paper we give the topological classification of real primary Kodaira surfaces and we describe in detail the structure of the corresponding moduli space. One of the main tools is the orbifold fundamental group of a real variety. Our first result is that if $(S, \sigma)$ is a real primary Kodaira 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 $(S, \sigma)$. Finally, we show that once we fix the topological type of $(S, \sigma)$ corresponding to a real primary Kodaira surface, the corresponding moduli space is irreducible (and connected).