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Annals of Global Analysis and Geometry

, Volume 51, Issue 4, pp 401–417 | Cite as

On toric locally conformally Kähler manifolds

  • Farid Madani
  • Andrei Moroianu
  • Mihaela Pilca
Article

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.

Keywords

Locally conformally Kähler structure Hopf surface Toric manifold Twisted Hamiltonian 

Mathematics Subject Classification

53A30 53B35 53C25 53C29 53C55 

Notes

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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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Institut für MathematikGoethe Universität FrankfurtFrankfurt am MainGermany
  2. 2.Laboratoire de Mathématiques de Versailles, UVSQ, CNRSUniversité Paris-SaclayVersaillesFrance
  3. 3.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  4. 4.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania

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