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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hard Lefschetz theorem for Vaisman manifolds

Authors: Beniamino Cappelletti-Montano, Antonio De Nicola, Juan Carlos Marrero and Ivan Yudin
Journal: Trans. Amer. Math. Soc. 371 (2019), 755-776
MSC (2010): Primary 53C25, 53C55, 53D35
Published electronically: October 17, 2018
MathSciNet review: 3885160
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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
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy

Antonio De Nicola
Affiliation: Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II 132, 84084 Fisciano, Italy

Juan Carlos Marrero
Affiliation: Unidad Asociada ULL-CSIC “Geometría Diferencial y Mecánica Geométrica” Departamento de Matemáticas, Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de La Laguna, La Laguna, Tenerife, Spain

Ivan Yudin
Affiliation: CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal

Keywords: Hard Lefschetz theorem, Vaisman manifolds, Sasakian manifolds, locally conformal symplectic manifolds, contact manifolds
Received by editor(s): July 20, 2016
Published electronically: October 17, 2018
Additional Notes: This work was partially supported by CMUC – UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020 (the second and fourth authors), by MICINN (Spain) and European Union (Feder) grants MTM2012-34478 and MTM 2015-64166-C2-2P (the second and third authors), by the European Community IRSES-project GEOMECH-246981 (the third author), by Prin 2015 – Varietà reali e complesse: geometria, topologia e analisi armonica – Italy, by the project GESTA funded by Fondazione di Sardegna and Regione Autonoma della Sardegna (the first author), and by the exploratory research project in the frame of Programa Investigador FCT IF/00016/2013 (the fourth author). The third author acknowledges the Centre for Mathematics of the University of Coimbra in Portugal for its support and hospitality during a visit where a part of this work was done.
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