Extension of a positive quasi-plurisuperharmonic current
This paper is concerned with the problem of extension of a positive current through a ``small'' obstacle. Namely, let \(A\) be a closed subset of an open set \(\Omega\subset\mathbb{C}^n\) and let \(T\) be a positive current of bidimenstion \((p,p)\) on \(\Omega\setminus A\). The problem is to give geometric conditions on the obstacle \(A\) and the current \(T\) which insure that the current \(T\) can be extended to positive current on \(\Omega\) with additional properties. This problem has its origin in the following remarkable result of \textit{E. Bishop} [Mich. Math. J. 11, 289--304 (1964; Zbl 0143.30302)]: if \(A\) is an proper analytic subset of \(\Omega\) and \(X\) is an analytic subset of \(\Omega\setminus A\) of pure dimension and locally finite volume near the points of \(A\), then the closure of \(X\) is an analytic subset of \(\Omega\). In [Invent. Math. 66, 361--376 (1982; Zbl 0488.58002)], \textit{H. Skoda} proved the following deep generalization of Bishop's result: if \(A\) is a (proper) analytic subset of \(\Omega\) and \(T\) is a closed positive current on \(\Omega\setminus A\) with locally finite mass near the points of \(A\), then its trivial extension \(\widetilde T:=\mathbf{1}_{\Omega\setminus A}T\) is a closed positive current. Later, \textit{H. El Mir} [Acta Math. 153, 1--45 (1984; Zbl 0557.32003)] gave an interesting generalization of Skoda's result to the case when the obstacle \(A\) is a locally complete pluripolar subset of \(\Omega\), i.e. locally \(A\) is given as the set where a plurisubharmonic function takes the value \(-\infty\). When the current is not assumed to be closed, \textit{T.-C. Dinh} and \textit{N. Sibony} [Manuscr. Math. 123, No. 3, 357--371 (2007; Zbl 1128.32020)] gave the following generalization of El Mir's theorem: if \(T\) is positive and quasi-plurisurharmonic (e.g. \(dd^cT\leq 0\)), then \(dd^cT\) has locally finite mass near the points of \(A\) and \(d\widetilde d^cT-dd^c\widetilde T\) is a positive current supported on \(A\). Many authors including the above ones obtained several extension results when the current \(T\) and the obstacle \(A\) satisfy more general conditions (see the references in the present paper). The first main result of this paper can be stated as follows: If \(A\) is a complete pluripolar set in \(\Omega\), \(T\) is a positive and quasi-plurisuperhamonic current of bidimension \((p,p)\) in \(\Omega\setminus A\) such that the \(2p\)-dimensional Hausdorff measure of \(\overline{\text{Supp\,}T}\cap A\) is \(0\), then its trivial extension \(\widetilde T\) is a positive quasi-plurisuperhamonic current on \(\Omega\) and \(d\widetilde d^cT- dd^c\widetilde T\) is a positive current supported on \(A\). The idea of the proof is to show that under the support assumption, the current \(T\) has locally finite mass and apply Dinh and Sibony's result. Other extension results are given for a positive and quasi-plurisuperharmonic current \(T\) of bidimension \((p,p)\) when the obstacle \(A\) satisfies one of the following conditions: 1. \(A\) is the zero set of a positive strictly \(k\)-convex function with \(k\leq p- 1\) (Theorem 3.5), 2. The \((2p-2)\)-Hausdorff measure is locally finite, 3. \(A\) is a non Levi-flat CR-submanifold with maximal ``Levi-flatness defect'' less than \(p-1\).
scientific article; zbMATH DE number 5629442