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On a class of Kato manifolds Otiman, Alexandra


In this talk we describe Kato manifolds, also known as manifolds with global spherical shell. We revisit Brunellaâ s proof of the fact that Katosurfaces admit locally conformally K\"ahler metrics, and we show that it holdsfor a large class of higher dimensional complex manifolds containing a globalspherical shell. On the other hand, we construct manifolds containing a globalspherical shell which admit no locally conformally K\"ahler metric. We then con-sider a specific class, which can be seen as a higher dimensional analogue of Inoue-Hirzebruch surfaces, and study several of their analytical properties. Inparticular, we give new examples, in any complex dimension $n\geq 3$, of com-pact non-exact locally conformally K\"ahler manifolds with algebraic dimension $n-2$, algebraic reduction bimeromorphic to $\C\Proj^{n-2}$ and admitting non-trivialholomorphic vector fields. These results are joint work with Nicolina Istrati(University of Tel Aviv) and Massimiliano Pontecorvo (Roma Tre University).

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