LcK structures with holomorphic Lee vector field on Vaisman-type manifolds

We give a complete description of all locally conformally K\"ahler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is not homothetic to the Lee vector field of a Vaisman st...

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Veröffentlicht in:arXiv.org 2019-05
Hauptverfasser: Madani, Farid, Moroianu, Andrei, Pilca, Mihaela
Format: Artikel
Sprache:eng
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Zusammenfassung:We give a complete description of all locally conformally K\"ahler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is not homothetic to the Lee vector field of a Vaisman structure. More generally, dropping the condition of being of Vaisman type, we show that on a compact complex manifold, any lcK metric with potential and with holomorphic Lee vector field admits a potential which is positive and invariant along the anti-Lee vector field.
ISSN:2331-8422
DOI:10.48550/arxiv.1905.07300