On a Class of Kato Manifolds

Abstract

Abstract We revisit Brunella’s proof of the fact that Kato surfaces admit locally conformally Kähler 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 that admit no locally conformally Kähler 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 Kähler manifolds with algebraic dimension $n-2$, algebraic reduction bimeromorphic to $\mathbb{C}\mathbb{P}^{n-2}$, and admitting nontrivial holomorphic vector fields.