The Kahler Rank of Compact Complex Manifolds

The Kahler rank was introduced by Harvey and Lawson in their 1983 paper as a measure of the kahlerianity of a compact complex surface. In this work we generalize this notion to the case of compact complex manifolds and we prove several results related to this notion. We show that on class VII surfaces, there is a correspondence between the closed positive forms on a surface and those on a blow-up in a point. We also show that a manifold of maximal Kahler rank which satisfies an additional condition is in fact Kahler.