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Note on pull-back and Lelong number of currents. (English) Zbl 0937.32005

Let \(f:(\mathbb{C}^m,0) \to(\mathbb{C}^n,0)\) be a holomorphic map. Let \(T\) be a positive closed current of bidegree (1,1) defined in a neighborhood of the origin in \((\mathbb{C}^n,0)\), and \(u\) be a plurisubharmonic potential for \(T\) such that \(T=dd^cu\).
The author formulates the following
Theorem 2. The following statements are equivalent:
1) the map \(f\) has maximal rank equal to \(n\); 2) \(f^*T:= dd^c(u\circ f)\) is well defined, and the operator \(f^*\) is continuous for the weak topology of currents;
3) the range of \(f\) is not pluripolar;
4) there exist a constant \(C>0\) (depending only on \(f)\) such that the inequality \[ \nu(T,0) \leq\nu (f^*T,0)\leq C_\nu (T,0) \] is valid where \(\nu (T,0)\) and \(\nu(f^*T,0)\) are the Lelong numbers of \(T\) and \(f^*T\) at the origin correspondingly.
The main result of the paper is the implication \(1)\Rightarrow 4)\).

MSC:

32C30 Integration on analytic sets and spaces, currents
32U25 Lelong numbers
32V05 CR structures, CR operators, and generalizations
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References:

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