Extension of Positive Currents with Special Properties of Monge-Ampère Operators

Abstract

In this paper we study the extension of currents across small obstacles. Our main results are: 1) Let $A$ be a closed complete pluripolar subset of an open subset $\Omega$ of $\mathsf{C}^n$ and $T$ be a negative current of bidimension $(p,p)$ on $\Omega\setminus A$ such that $dd^{c}T\geq-S$ on $\Omega\setminus A$ for some positive plurisubharmonic current $S$ on $\Omega$. Assume that the Hausdorff measure $\mathscr{H}_{2p}(A\cap \overline{\operatorname{Supp} T})=0$. Then $\widetilde{T}$ exists. Furthermore, the current $R= \widetilde{dd^{c}T}-{dd}^{c} \widetilde{T}$ is negative supported in $A$. 2) Let $u$ be a positive strictly $k$-convex function on an open subset $\Omega$ of $\mathsf{C}^n$ and set $A=\{z\in\Omega:u(z)=0\}$. Let $T$ be a negative current of bidimension $(p,p)$ on $\Omega\setminus A$ such that $dd^{c}T\geq -S$ on $\Omega\setminus A$ for some positive plurisubharmonic (or $dd^{c}$-negative) current $S$ on $\Omega$. If $p\geq k+1$, then $\widetilde{T}$ exists. If $p\geq k+2$, $dd^{c}S\leq 0$ and $u$ of class $\mathscr{C}^{2}$, then $\widetilde{dd^{c}T}$ exists and $\widetilde{dd^{c}T}= dd^{c}\widetilde{T}$.