On compact complex surfaces of Kähler rank one

Details On compact complex surfaces of Kähler rank one On compact complex surfaces of Kähler rank one

2010-10-13

arxiv.org/abs/1010.2591v1

Mathematics

Complex Variables

Submitted to Amer. J. Math

Scientific paper

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.

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