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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 193, Issue 5, pp 1345–1351 | Cite as

Holomorphic submersions of locally conformally Kähler manifolds

  • Liviu Ornea
  • Maurizio Parton
  • Victor Vuletescu
Article

Abstract

A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold \(X\) admits a holomorphic submersion with positive-dimensional fibers at least one of which is of Kähler type, then \(X\) is globally conformally Kähler or biholomorphic, up to finite covers, to a small deformation of a Vaisman manifold (i.e., a mapping torus over a circle, with Sasakian fiber). As a consequence, we show that the product of a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.

Keywords

Locally conformally Kähler manifold Holomorphic submersion Vaisman manifold 

Mathematics Subject Classification (2010)

53C55 

Notes

Acknowledgments

Liviu Ornea and Victor Vuletescu are partially supported by CNCS UEFISCDI, Project Number PN-II-ID-PCE-2011-3-0118. All authors thank the anonymous referee for carefully reading their paper and for his very useful comments.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Liviu Ornea
    • 1
    • 2
  • Maurizio Parton
    • 3
  • Victor Vuletescu
    • 1
  1. 1.Faculty of MathematicsUniversity of BucharestBucharestRomania
  2. 2.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  3. 3.Dipartimento di ScienzeUniversita di Chieti—PescaraPescaraItaly

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