Mathematische Zeitschrift

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

Examples of non-trivial rank in locally conformal Kähler geometry

  • Maurizio Parton
  • Victor Vuletescu


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 r M of a locally conformal Kähler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that r M 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 r M = b 1 or r M = b 1/2.


Complex Manifold Galois Group Betti Number Maximal Rank Galois Extension 
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  1. 1.
    Belgun, F., Moroianu, A.: Holonomy of Tame Weyl Structures. electronic preprint, July 2009. arXiv: 0907.3182v1Google Scholar
  2. 2.
    Gini, R., Ornea, L., Parton, M.: Locally conformal Kähler reduction. J. Reine Angew. Math. 581 (April 2005)Google Scholar
  3. 3.
    Gini R., Ornea L., Parton M., Piccinni P.: Reduction of Vaisman structures in complex and quaternionic geometry. J. Geom. Phys. 56(12), 2501–2522 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Milne, J.S.: Fields and Galois Theory, September 2008., version 4.21
  5. 5.
    Milne, J.S.: Algebraic Number Theory, April 2009., version 3.02
  6. 6.
    Oeljeklaus K., Toma M.: Non-Kähler compact complex manifolds associated to number fields. Ann. Inst. Fourier 55(1), 1291–1300 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Ornea L., Verbitsky M.: Morse–Novikov cohomology of locally conformally Kähler manifolds. J. Geom. Phys. 59(3), 295–305 (2009)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Ornea, L., Verbitsky, M.: Locally conformal Kähler manifolds with potential. Math. Ann. (2010). doi: 10.1007/s00208-009-0463-0
  9. 9.
    Vaisman I.: On locally conformal almost Kähler manifolds. Israel J. Math. 24, 338–351 (1976)CrossRefMATHMathSciNetGoogle Scholar

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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