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arXiv · 1510.04946

Hard Lefschetz Theorem for Vaisman manifolds

Beniamino Cappelletti-Montano·Antonio De Nicola·Juan Carlos Marrero·Ivan Yudin
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Abstract

We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual $1$-form coincides with the Lee $1$-form. Finally, we discuss several examples of compact l.c.s. manifolds of the first kind which do not admit compatible Vaisman metrics.

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Beniamino Cappelletti-Montano, Antonio De Nicola, Juan Carlos Marrero, Ivan Yudin. 2016-07-20. Hard Lefschetz Theorem for Vaisman manifolds. https://doi.org/10.1090/tran%2F7525

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