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 Kahler 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.

中文翻译：

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一类 Kato 歧管

我们重新审视了布鲁内拉的证明，即加藤表面局部共形地承认 K\" ahler 度量，我们证明它适用于一大类包含全局球壳的高维复杂流形。另一方面，我们构造了包含一个不允许局部共形 Kahler 度量的全局球壳。我们考虑这些流形的特定类别，可以将其视为 Inoue-Hirzebruch 曲面的更高维类似物，并研究它们的几个分析性质。特别是，我们给出了新的例子，在任何复维数 $n \geq 3$ 中，具有代数维数 $n-2$ 的紧凑非精确局部保形 K\" ahler 流形，代数约简双同胚 $\mathbb{C}\mathbb{P}^{ n-2}$ 并承认非平凡的全息向量场。