Mathematische Zeitschrift

, Volume 270, Issue 1–2, pp 179–187 | Cite as

Examples of non-trivial rank in locally conformal Kähler geometry

Article

Abstract

We consider locally conformal Kähler geometry as an equivariant, homothetic Kähler geometry (K, Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γ to its dilation factors, thus completing the description of locally conformal Kähler geometry in this equivariant setting. The rank rM of a locally conformal Kähler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that rM is non-trivial, providing explicit examples of locally conformal Kähler manifolds with \({1\nless{\text{\upshape \rmfamily r}_{M}}\nless b_1}\). As far as we know, these are the first examples of this kind. Moreover, we prove that locally conformal Kähler Oeljeklaus-Toma manifolds have either rM = b1 or rM = b1/2.

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di ScienzeUniversità di Chieti-PescaraPescaraItaly
  2. 2.Faculty of Mathematics and InformaticsUniversity of BucharestBucharestRomania

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