Existence criteria for special locally conformally Kähler metrics
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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 (M, J) 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 (M, J) admits an LCK metric with positive potential. Finally, we obtain new non-existence results for LCK metrics on certain products of complex manifolds.
KeywordsLocally conformally Kähler metric Vaisman metric Torus action Lee field
Mathematics Subject Classification53A30 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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