Metric properties of manifolds bimeromorphic to compact Kähler spaces

It is known that if ˜M is a modification of a compact complex manifold M, and if M is Kähler, in general ˜M fails to be Kähler; conversely, if ˜M is Kähler, in general M fails to be Kähler also in the case of a blow-up with smooth center. The authors prove that if ˜M is balanced and satisfies (B) (in particular, ˜M is Kähler; here (B) is a cohomology condition), then M is balanced and satisfies (B). Previously, the authors have shown that if M is balanced, then ˜M is balanced. As a consequence, every manifold in the class of Fujiki is balanced, and every Moishezon manifold is balanced. The proof is based on the following theorem: Suppose M and ˜M are complex manifolds (not necessarily compact) and f: ˜M to M is a proper modification. If T is a positive (de-debar)-closed current on M of degree (1, 1), then there exists a positive (de-debar)-closed current ˜T on ˜M of degree (1, 1) such that f ˜T = T. Moreover, if ˜M is compact, such a current is unique.