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Transforms of currents by modifications and 1-convex manifolds. (English) Zbl 1034.32009
The following question is discussed. Let $$\pi:\widetilde X\to X$$ be a proper modification of complex manifolds with center $$Z$$ and exceptational divisor $$E$$. In analogy with the definition of a strict transform of a hypersurface in $$X$$, the authors consider the case of currents on $$X$$. They give nice results for currents of bidegree $$(1,1)$$. As an application a result by the reviewer [ibid. 38, No. 2, 287–294 (2001; Zbl 0982.32010)] on the equivalence between embeddability and Kählerianity of certain 1-convex manifolds is generalized.

##### MSC:
 32F10 $$q$$-convexity, $$q$$-concavity 32E05 Holomorphically convex complex spaces, reduction theory
##### Keywords:
1-convexity; $$(1,1)$$-currents