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Extension of plurisubharmonic currents. (English) Zbl 1046.32009

Summary: Let \(A\) be a closed subset of an open subset \(\Omega\) of \(\mathbb{C}^n\) and \(T\) be a negative current on \(\Omega\setminus A\) of bidimension \((p,p)\). Assume that \(T\) is psh and \(A\) is complete pluripolar such that the Hausdorff measure \({\mathcal H}_{2p}(\overline {\text{Supp}\,T}\cap A)=0\), then \(T\) extends to a negative psh current on \(\Omega\). We also show that if \(T\) is psh or if \(dd^cT\) extends to a current with locally finite mass on \(\Omega\), then the trivial extension \(\widetilde T\) of \(T\) by zero across \(A\) exists in both cases: \(A\) is the zero set of a \(k\)-convex function with \(k\leq p-1\) or \({\mathcal H}_{2(p-1)} (\overline{\text{Supp}\,T}\cap A)=0\).
Our basic tool is the following theorem: Let \(A\) be a closed complete pluripolar subset of an open subset \(\Omega\) of \(\mathbb{C}^n\) and \(T\) be a positive current of bidimension \((p,p)\) on \(\Omega\setminus A\). Suppose that \(\widetilde T\) and \(\widetilde {dd^cT}\) exist (resp. \(\widetilde T\) exists and \(dd^c T\leq 0\) on \(\Omega\setminus A)\), then there exists a positive (resp. closed positive) current \(S\) supported in \(A\) such that \(\widetilde {dd^cT}=dd^c\widetilde T+S\).
Furthermore, we give a generalization of some theorems done by Siu and Ben Messaoud-El Mir and Alessandrini-Bassanelli without requiring anything from \(dT\).

MSC:

32U40 Currents
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