On the structure of co-Kähler manifolds

Abstract

By the work of Li, a compact co-Kähler manifold $$M$$ is a mapping torus $$K_\varphi $$ , where $$K$$ is a Kähler manifold and $$\varphi $$ is a Hermitian isometry. We show here that there is always a finite cyclic cover $$\overline{M}$$ of the form $$\overline{M} \cong K \times S^1$$ , where $$\cong $$ is equivariant diffeomorphism with respect to an action of $$S^1$$ on $$M$$ and the action of $$S^1$$ on $$K \times S^1$$ by translation on the second factor. Furthermore, the covering transformations act diagonally on $$S^1, K$$ and are translations on the $$S^1$$ factor. In this way, we see that, up to a finite cover, all compact co-Kähler manifolds arise as the product of a Kähler manifold and a circle.