On the structure of co-Kähler manifolds

By the work of Li, a compact co-Kähler manifold $M$ is a mapping torus $K_φ$, where $K$ is a Kähler manifold and $φ$ is a Hermitian isometry. We show here that there is always a finite cyclic cover $\bar M$ of the form $\bar M \cong K \times S^1$, 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.

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