The automorphism group of Kodaira surfaces

In this paper we give an explicit description of the automorphism group of a primary Kodaira surface $X$ in terms of suitable liftings to the universal cover $\mathbb{C}^2$. As it happens for complex tori, the automorphism group of $X$ is an extension of a finite cyclic group related to the action of the automorphisms on the tangent bundle of $X$ by a group of automorphisms which acts on $X$ as translations (both on the base and on the fibres of the elliptic fibraion of $X$). We can then characterize those automorphisms acting symplectically and those acting trivially on cohomology. Finally, we describe the fixed locus of an automorphism and we show that, if not empty, it is the union of a finite number of fibres of the elliptic fibration of $X$. As a byproduct, we provide a different proof of Borcea's description of the moduli space of Kodaira surfaces.