LcK structures with holomorphic Lee vector field on Vaisman-type manifolds
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LcK structures with holomorphic Lee vector field on Vaisman-type manifolds Farid Madani1 · Andrei Moroianu2
· Mihaela Pilca1
Received: 28 February 2020 / Accepted: 22 October 2020 © Springer Nature B.V. 2020
Abstract We give a complete description of all locally conformally Kähler 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. Keywords Locally conformally Kähler structure · Holomorphic Lee vector field · Vaisman structure · LcK structure with potential Mathematics Subject Classification (2010) 53A30 · 53B35 · 53C25 · 53C29 · 53C55
1 Introduction A Hermitian metric g on a complex manifold (M, J ) is called locally conformally Kähler (in short, lcK) if around any point in M, the metric g can be conformally rescaled to a Kähler metric. This condition is equivalent to the existence of a closed 1-form θ such that d = θ ∧ , where denotes the fundamental 2-form defined as (·, ·) := g(J ·, ·). The 1-form θ is called the Lee form and its metric dual is called the Lee vector field. In this paper we assume that θ = 0, i.e. (J , g, ) is not Kähler. A special class of lcK structures is represented by the so-called Vaisman structures, defined by the property that the Lee form is non-zero and parallel with respect to the Levi-Civita
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Andrei Moroianu [email protected] Farid Madani [email protected] Mihaela Pilca [email protected]
1
Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040 Regensburg, Germany
2
CNRS, Laboratoire de mathématiques d’Orsay, Université Paris-Saclay, 91405 Orsay, France
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Geometriae Dedicata
connection of g. It is known that a Vaisman structure on a complex manifold is uniquely determined, up to a positive constant, by its Lee form, via the following identity: =
1 (θ ∧ J θ − d J θ ). |θ |2
On a compact complex manifold of Vaisman type, i.e. admitting at least one Vaisman metric, the Lee vector fields of all Vaisman structures are holomorphic, and coincide up to a positive multiplicative constant. This fact was originally obtained in [12], but for the reader’s convenience we give below an alternative proof. In [8], Moroianu et al. proved that a compact lcK manifold with holomorphic Lee vector field is Vaisman if the Lee vector field either has constant norm or is divergence-free. Moreover, they also construct examples of non-Vaisman lcK structures with holomorphic Lee vector field on a compact manifold of Vaisman type. These lcK structures, however, have the same Lee vector field as the Vaisman structures. More recently, Belgun [1] constructed examples of lc
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