By the work of Li, a compact co-Kähler manifold M is a mapping torus Kφ, where K is a Kähler manifold and φ is a Hermitian isometry. We show here that there is always a finite cyclic cover M¯¯¯¯¯ of the form M¯¯¯¯¯≅K×S1, where ≅ is equivariant diffeomorphism with respect to an action of S1 on M and the action of S1 on K×S1 by translation on the second factor. Furthermore, the covering transformations act diagonally on S1,K and are translations on the S1 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.
Bazzoni, Giovanni and Oprea, John, "On The Structure of Co-Kähler Manifolds" (2014). Mathematics Faculty Publications. 211.
The final publication is available at Springer via http://dx.doi.org/10.1007/s10711-013-9869-7