On the structure of co-Kähler manifolds
Abstract
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
M
¯
of the form
M
¯
≅K×
S
1
, where
≅
is equivariant diffeomorphism with respect to an action of
S
1
on
M
and the action of
S
1
on
K×
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.