Existence criteria for special locally conformally Kähler metrics

  • Nicolina Istrati


We investigate the relation between holomorphic torus actions on complex manifolds of locally conformally Kähler (LCK) type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold (MJ) contains a compact torus which is not totally real, then there exists a Vaisman metric on the manifold, generalising a result of Kamishima–Ornea. Also, we obtain a new obstruction to the existence of LCK structures on a given complex manifold in terms of its automorphism group. As an application, we obtain a classification of manifolds of LCK type among all the manifolds having the structure of a holomorphic principal torus bundle. Moreover, we show that if the group of biholomorphisms contains a compact torus whose dimension is half the real dimension of M, then (MJ) admits an LCK metric with positive potential. Finally, we obtain new non-existence results for LCK metrics on certain products of complex manifolds.


Locally conformally Kähler metric Vaisman metric Torus action Lee field 

Mathematics Subject Classification

53A30 53C25 53B35 



I am grateful to Andrei Moroianu for his encouragement and valuable suggestions that improved this paper. Also I thank Paul Gauduchon for pointing out an error in a preliminary version of the paper.


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018
corrected publication [August/2018]

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu - Paris Rive Gauche, UFR de MathémathiquesUniversité Paris Diderot - Paris 7Paris Cedex 13France

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