On the structure of co-Kähler manifolds

Abstract

By the work of Li, a compact co-Kähler manifold $M$ is a mapping torus $K_{\phi }$, where $K$ is a Kähler manifold and $\phi$ 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.

20 pages; revised version: new results on fundamental group of co-Kähler manifolds and on compact co-Kähler manifolds which are not products. To appear in Geom. Dedicata

References in corpus (1)

• Aspherical Kähler Manifolds with Solvable Fundamental Group