Mathematische Annalen

, Volume 317, Issue 1, pp 1–40

On the metric structure of non-Kähler complex surfaces

  • Florin Alexandru Belgun
Original article

DOI: 10.1007/s002080050357

Cite this article as:
Belgun, F. Math Ann (2000) 317: 1. doi:10.1007/s002080050357


We give a characterization of a locally conformally Kähler (l.c.K.) metric with parallel Lee form on a compact complex surface. Using the Kodaira classification of surfaces, we classify the compact complex surfaces admitting such structures. This gives a classification of Sasakian structures on compact three-manifolds. A weak version of the above mentioned characterization leads to an explicit construction of l.c.K. metrics on all Hopf surfaces. We characterize the locally homogeneous l.c.K. metrics on geometric complex surfaces, and we prove that some Inoue surfaces do not admit any l.c.K. metric.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Florin Alexandru Belgun
    • 1
  1. 1.Centre de Mathématiques, UMR 7640 CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France (e-mail:

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