The moduli space of Keum–Naie surfaces

Abstract

Using a new description of the surfaces discovered by Keum and later investigated by Naie, and of their fundamental group, we prove the following main result.

Let $S$ be a smooth complex projective surface which is homotopically equivalent to a Keum–Naie surface. Then $S$ is a Keum–Naie surface. The connected component of the Gieseker moduli space corresponding to Keum–Naie surfaces is irreducible, normal, unirational of dimension 6.