Annali di Matematica Pura ed Applicata (1923 -)

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

Holomorphic submersions of locally conformally Kähler manifolds

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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 

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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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