# On a class of Kato manifolds

## TL;DR

This paper explores the existence of locally conformally Kähler metrics on higher-dimensional complex manifolds with a global spherical shell, providing new examples and analyzing their geometric and analytical properties.

## Contribution

It extends Brunella's results to higher dimensions, constructs new manifolds with spherical shells that lack such metrics, and studies their complex geometric features.

## Key findings

- Brunella's proof applies to a broad class of higher-dimensional manifolds.
- Constructed manifolds with spherical shells that do not admit LCK metrics.
- Provided examples of non-exact LCK manifolds with specific algebraic properties.

## Abstract

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.