Remarks on a theorem of Cupit-Foutou and Zaffran
Commun. Korean Math. Soc.
Published online November 12, 2019
Jinhong Kim
Chosun University
Abstract : There is a well-known class of compact, complex, non-K\" ahlerian manifolds constructed by Bosio, called the LVMB manifolds, which properly includes the Hopf manifold, the Calabi-Eckmann manifold, and the LVM manifolds. As in the case of LVM manifolds, these LVMB manifolds can admit a regular holomorphic foliation $\mathcal{F}$. Moreover, later Meersseman showed that if an LVMB manifold is actually an LVM manifold, then the regular holomorphic foliation $\mathcal{F}$ is actually transverse K\" ahler. The aim of this paper is to deal with a converse question. That is, we show that, when the holomorphic foliation $\mathcal{F}$ on an LVMB manifold $N$ is transverse K\" ahler with respect to a basic and transverse K\" ahler form and the leaf space $N/\mathcal{F}$ is an orbifold, $N/\mathcal{F}$ is projective, and thus $N$ is actually an LVM manifold. This reproves a well-known result of Cupit-Foutou and Zaffran.
Keywords : LVM manifolds, LVMB manifolds, holomorphic foliations, transverse K\" ahler
MSC numbers : 53D20
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