On compact complex surfaces of K\"ahler rank one

Ionut Chiose; Matei Toma

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arXiv · 1010.2591

On compact complex surfaces of K\"ahler rank one

Ionut Chiose·Matei Toma

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Abstract

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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Ionut Chiose, Matei Toma. 2010-10-13. On compact complex surfaces of K\"ahler rank one. https://arxiv.org/abs/1010.2591

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