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Positive \(\partial\bar\partial\)-closed currents and non-Kähler geometry. (English) Zbl 0735.32008
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.

MSC:
32C30 Integration on analytic sets and spaces, currents
32Q15 Kähler manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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