The K\"ahler rank of compact complex manifolds

The K\"ahler rank was introduced by Harvey and Lawson in their 1983 paper as a measure of the {\it k\"ahlerianity} 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 K\"ahler rank which satisfies an additional condition is in fact K\"ahler.