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arXiv · 1206.1981

Holomorphic submersions of locally conformally K\"ahler manifolds

Liviu Ornea·Maurizio Parton·Victor Vuletescu
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Abstract

A locally conformally K\"ahler (LCK) manifold is a complex manifold covered by a K\"ahler 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 K\"ahler type, then X is globally conformally K\"ahler or biholomorphic, up to finite covers, to a Vaisman manifold (i.e. a mapping torus over a circle, with Sasakian fibre). As a consequence, we show that the product between a compact non-K\"ahler LCK and a compact K\"ahler manifold cannot carry a LCK metric.

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Liviu Ornea, Maurizio Parton, Victor Vuletescu. 2012-06-09. Holomorphic submersions of locally conformally K\"ahler manifolds. https://arxiv.org/abs/1206.1981

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