The K\"ahler rank of compact complex manifolds

TL;DR

This paper generalizes the K"ahler rank concept from surfaces to higher-dimensional compact complex manifolds, establishing new relations and conditions under which manifolds are K"ahler.

Contribution

It extends the notion of K"ahler rank to compact complex manifolds and proves new results linking positive forms and K"ahler properties.

Findings

Correspondence between positive forms on class VII surfaces and their blow-ups

Maximal K"ahler rank plus an additional condition implies the manifold is K"ahler

Generalization of K"ahler rank to higher dimensions

Abstract

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.