Holomorphic submersions of locally conformally Kähler manifolds
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A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold \(X\) admits a holomorphic submersion with positive-dimensional fibers at least one of which is of Kähler type, then \(X\) is globally conformally Kähler or biholomorphic, up to finite covers, to a small deformation of a Vaisman manifold (i.e., a mapping torus over a circle, with Sasakian fiber). As a consequence, we show that the product of a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.
KeywordsLocally conformally Kähler manifold Holomorphic submersion Vaisman manifold
Mathematics Subject Classification (2010)53C55
Liviu Ornea and Victor Vuletescu are partially supported by CNCS UEFISCDI, Project Number PN-II-ID-PCE-2011-3-0118. All authors thank the anonymous referee for carefully reading their paper and for his very useful comments.
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