On Locally Conformally Kähler Threefolds with Algebraic Dimension Two

The paper is part of an attempt of understanding non-Kähler threefolds. We start by looking at compact complex non-Kähler threefolds with algebraic dimension two and admitting lcK metrics. Under certain assumptions, we prove that they are blown-up quasi-bundles over a projective surface.

On a Class of Kato Manifolds

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.

Locally conformally Kähler structures on four-dimensional solvable Lie algebras

We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the 4-dimensional structures in our classification.

HOMOGENEOUS SASAKI AND VAISMAN MANIFOLDS OF UNIMODULAR LIE GROUPS

A Vaisman manifold is a special kind of locally conformally Kähler manifold, which is closely related to a Sasaki manifold. In this paper, we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete classification of simply connected Sasaki and Vaisman unimodular Lie groups, up to modification.