Extension of Positive Currents with Special Properties of Monge-Ampère Operators - Abdulaali, Ahmad L. al (MATHEMATICA SCANDINAVICA, 2013)
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 Ω of C^n and T be a negative current of bidimension (p,p) on ΩA such that dd^cT≥-S on ΩA for some positive plurisubharmonic current S on Ω. Assume that the Hausdorff measure H_2p(ASupp T)=0. Then T exists. Furthermore, the current R= dd^cT-dd^c T is negative supported in A. 2) Let u be a positive strictly k-convex function on an open subset Ω of C^n and set A={zΩ:u(z)=0}. Let T be a negative current of bidimension (p,p) on ΩA such that dd^cT≥ -S on ΩA for some positive plurisubharmonic (or dd^c-negative) current S on Ω. If p≥ k+1, then T exists. If p≥ k+2, dd^cS≤ 0 and u of class C^2, then dd^cT exists and dd^cT= dd^cT.