On compact complex surfaces of K\"ahler rank one

The K"ahler rank of compact complex surfaces was introduced by Harvey and Lawson in their 1983 paper on K"ahler manifolds as a measure of ``k"alerianity’'. Here we give a partial classification of compact complex surfaces of K"ahler 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 K"ahler rank is a birational invariant.