Positive ddbar-closed currents and non-Kähler geometry

In this paper some new results on positive (de-debar)−closed currents are applied to modifications of compact complex manifolds. The main result in this topic is that every smooth proper modification of a compact Kähler manifold M is balanced. Moreover, under suitable hypotheses on the map, the Kähler degrees of M corresponds to homological properties of the exceptional set of the modification. More examples of p-Kähler manifolds are discussed in the last section of the paper.