  We revisit Brunella's proof of the fact that Kato surfaces admit locally
conformally K\" ahler metrics, and we show that it holds for a large class of
higher dimensional complex manifolds containing a global spherical shell. On
the other hand, we construct manifolds containing a global spherical shell
which admit no locally conformally K\"ahler metric. We consider a specific
class of these manifolds, which can be seen as a higher dimensional analogue of
Inoue-Hirzebruch surfaces, and study several of their analytical properties. In
particular, we give new examples, in any complex dimension $n \geq 3$, of
compact non-exact locally conformally K\" ahler manifolds with algebraic
dimension $n-2$, algebraic reduction bimeromorphic to
$\mathbb{C}\mathbb{P}^{n-2}$ and admitting non-trivial holomorhic vector
fields.
