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On toric locally conformally Kähler manifolds

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Abstract

We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is \(-\infty \), and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.

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Acknowledgements

This work was supported by the Procope Project No. 32977YJ and by the SFB 1085. We thank Nicolina Istrati for pointing out to us an error in a preliminary version of the paper and for useful suggestions.

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Authors and Affiliations

  1. Institut für Mathematik, Goethe Universität Frankfurt, Robert-Mayer-Str. 10, 60325, Frankfurt am Main, Germany

    Farid Madani

  2. Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035, Versailles, France

    Andrei Moroianu

  3. Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040, Regensburg, Germany

    Mihaela Pilca

  4. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Grivitei Str., 010702, Bucharest, Romania

    Mihaela Pilca

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  1. Farid Madani
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  2. Andrei Moroianu
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  3. Mihaela Pilca
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Correspondence to Andrei Moroianu.

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Madani, F., Moroianu, A. & Pilca, M. On toric locally conformally Kähler manifolds. Ann Glob Anal Geom 51, 401–417 (2017). https://doi.org/10.1007/s10455-017-9545-5

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