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On construction of positive closed currents with prescribed Lelong numbers. (English) Zbl 07334094

Summary: We establish that a sequence \((X_k)_{k\in\mathbb{N}}\) of analytic subsets of a domain \(\Omega\) in \(\mathbb{C}^n\), purely dimensioned, can be released as the family of upper-level sets for the Lelong numbers of some positive closed current. This holds whenever the sequence \((X_k)_{k\in\mathbb{N}}\) satisfies, for any compact subset \(L\) of \(\Omega \), the growth condition \(\sum\limits_{k\in\mathbb{N}}C_k \operatorname{mes}(X_k\cap L)<\infty \). More precisely, we built a positive closed current \(\Theta\) of bidimension \((p,p)\) on \(\Omega \), such that the generic Lelong number \(m_{X_k}\) of \(\Theta\) along each \(X_k\) satisfies \(m_{X_k}=C_k\). In particular, we prove the existence of a plurisubharmonic function \(v\) on \(\Omega\) such that, each \(X_k\) is contained in the upper-level set \(E_{C_k}(dd^cv)\).

MSC:

32Bxx Local analytic geometry
32Cxx Analytic spaces
32Uxx Pluripotential theory
58Axx General theory of differentiable manifolds
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References:

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