Real Kodaira surfaces

TL;DR

This paper classifies real primary Kodaira surfaces topologically, describes their moduli space structure, and shows the orbifold fundamental group determines their differentiable types and topological classification.

Contribution

It provides a complete topological classification of real primary Kodaira surfaces using orbifold fundamental groups and describes the structure of their moduli space.

Findings

Differentiable type determined by orbifold fundamental group sequence

All topological types of real primary Kodaira surfaces identified

Moduli space for fixed topological type is irreducible and connected

Abstract

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).