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Zbl 1241.32028
Ghiloufi, Noureddine; Dabbek, Khalifa
Extension of a positive quasi-plurisuperharmonic current. (Prolongement d'un courant positif quasi-plurisurharmonique.) (French)
[J] Ann. Math. Blaise Pascal 16, No. 2, 287-304 (2009). ISSN 1259-1734

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 $\Omega\subset\bbfC^n$ 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.\par This problem has its origin in the following remarkable result of {\it 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 $\Omega$. \par In [Invent. Math. 66, 361--376 (1982; Zbl 0488.58002)], {\it 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 $\widetilde T:=\text{\bf 1}_{\Omega\setminus A}T$ is a closed positive current.\par Later, {\it 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 $-\infty$. \par When the current is not assumed to be closed, {\it T.-C. Dinh} and {\it 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. $dd^cT\le 0$), 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 $A$. \par 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). \par 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.\par 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:\par 1. $A$ is the zero set of a positive strictly $k$-convex function with $k\le p- 1$ (Theorem 3.5),\par 2. The $(2p-2)$-Hausdorff measure is locally finite,\par 3. $A$ is a non Levi-flat CR-submanifold with maximal ``Levi-flatness defect'' less than $p-1$.
[Ahmed Zeriahi (Toulouse)]

MSC 2000:: *32U05 Plurisubharmonic functions and generalizations
32U40 Currents
32V20 Analysis on CR manifolds

Keywords: plurisubharmonic function; pluripolar sets; positive current; extension; slicing

Citations: Zbl 0143.30302; Zbl 0488.58002; Zbl 0557.32003; Zbl 1128.32020

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