Positive \(\partial\bar\partial\)-closed currents and non-Kähler geometry (Q811498)

Some new results on positive \(\partial\bar\partial\)-closed currents (in particular a Support Theorem) are applied to modifications \(f:\tilde M\to M\). The main result in this topic is that if \(f:\tilde M\to M\) is a modification, \(M\) and \(\tilde M\) are compact complex manifolds and \(M\) is Kähler, then \(\tilde M\) is balanced. Moreover, under suitable hypotheses on the map, the Kähler degrees of \(\tilde M\) 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.