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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.
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참고문헌 (23)
참고문헌 신청
W. L. Baily / 1957 / On the imbedding of V -manifolds in projective space / Amer. J. Math. 79:403~430
F. Battaglia / 2001 / Generalized toric varieties for simple nonrational convex polytopes / Internat. Math. Res. Notices 2001(24):1315~1337
F. Battaglia / 2015 / Foliations modeling nonrational simplicial toric varieties / Int. Math. Res. Not. IMRN 2015(22):11785~11815
L. Battisti / 2013 / LVMB manifolds and quotients of toric varieties / Math. Z 275(1-2):549~568
F. Bosio / 2001 / Vari´et´es complexes compactes : une g´en´eralisation de la construction de Meersseman et L´opez de Medrano-Verjovsky / Ann. Inst. Fourier(Grenoble) 51(5):1259~1297
F. Battaglia / 2001 / Generalized toric varieties for simple nonrational convex polytopes / Internat. Math. Res. Notices 2001(24):1315~1337
F. Battaglia / 2015 / Foliations modeling nonrational simplicial toric varieties / Int. Math. Res. Not. IMRN 2015(22):11785~11815
L. Battisti / 2013 / LVMB manifolds and quotients of toric varieties / Math. Z 275(1-2):549~568
F. Bosio / 2001 / Vari´et´es complexes compactes : une g´en´eralisation de la construction de Meersseman et L´opez de Medrano-Verjovsky / Ann. Inst. Fourier(Grenoble) 51(5):1259~1297
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