Examples of non-trivial rank in locally conformal Kähler geometry

Details Examples of non-trivial rank in locally conformal Kähler geometry Examples of non-trivial rank in locally conformal Kähler geometry

2010-01-27

arxiv.org/abs/1001.4891v3

Mathematische Zeitschrift, Volume 270, Issue 1 (2012), Page 179-187

Mathematics

Differential Geometry

14 pages. v3: a minor change in proof of Theorem 5.1, updated Acknowledgments and Bibliography

Scientific paper

10.1007/s00209-010-0791-5

We consider locally conformal Kaehler geometry as an equivariant, homothetic Kaehler geometry (K,\Gamma). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting \Gamma to its dilation factors, thus completing the description of locally conformal Kaehler geometry in this equivariant setting. The rank r of a locally conformal Kaehler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that r is non-trivial, providing explicit examples of locally conformal Kaehler manifolds with 1

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