On a class of Kato manifolds

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 $n\ge 3$, 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.