On compact complex surfaces of Kähler rank one

The Kähler rank of compact complex surfaces was introduced by Harvey and Lawson in their 1983 paper on Kähler manifolds as a measure of ``kälerianity''. 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 birational invariant.

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