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On compact complex surfaces of Kähler rank one
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 135, Number 3, June 2013
- pp. 851-860
- 10.1353/ajm.2013.0022
- Article
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Abstract
The Kähler rank of compact complex surfaces was introduced by Harvey and Lawson in their 1983 paper on Kähler manifolds as a measure of “kählerianity”. Here we give a partial classification of compact complex surfaces of Kähler rank 1. These are either elliptic surfaces, or Hopf surfaces, or they admit a holomorphic foliation of a very special type. As a consequence we give an affirmative answer to the question raised by Harvey and Lawson whether the Kähler rank is a bimeromorphic invariant.
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