Communications of the
Korean Mathematical Society CKMS

There is a well-known class of compact, complex, non-K\"ahl\-erian 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 and to give a simple and new proof of a well-known result of Cupit-Foutou and Zaffran. 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.

Keywords: LVM manifolds, LVMB manifolds, holomorphic foliations, transverse K\" ahler

MSC numbers: 53D20, 14M25, 32M05, 57S25

Supported by: This study
was supported by research fund from Chosun University, 2019.