A note on the automorphism group of Kodaira surfaces

Abstract

We give an explicit description of the automorphism groups of primary and secondary Kodaira surfaces. We first prove that every automorphism of a Kodaira surface lifts to an affine transformation of the universal cover $\mathbb{C}^2$, and characterize precisely the affine transformations arising in this way for primary Kodaira surfaces. This allows us to compute explicitly the normalizer of the fundamental group and, consequently, to determine both the connected component of the identity and the group of connected components of the automorphism group. Passing to the canonical cover, we then obtain an analogous explicit description for secondary Kodaira surfaces, including a complete determination of the possible groups of connected components.