The extendability of \(S\)-plurisubharmonic currents (Q1931822)

The author proves the following result: If \(A\) is a closed pluripolar subset of an open subset \(\Omega\) of \(\mathbb C^{n}\) and \(T\) is a positive \(S\) plurisuperharmonic current of bidegree \((p,p)\) on \(\Omega\setminus A\) such that the \(2p-1\) Hausdorff measure \(\mathcal H_{2p-1}(A\cap\overline{\text{Supp T}})\) is zero, then one can extend this current through \(A\) to a current on the whole \(\Omega\) which moreover obeys the property that the extension of \(dd^{c} T\), less the \(dd^{c}\) of the extension, is positive and supported in \(A\). \(S\) plurisuperharmonicity means that the \(dd^c\) of this current is dominated by some positive current \(S\) of bidegree \((p+1,p+1)\) (note the misprint in the paper). The findings of this paper can be seen as a generalization of a classical result by Harvey.