Resumen de Extension of plurisubharmonic currents

Fredj Elkhadhra, Hassine El Mir, Khalifa Dabbek

• Let A be a closed subset of an open subset Ω of ℂn and T be a negative current on Ω\ A of bidimension (p,p). Assume that T is psh and A is complete pluripolar such that the Hausdorff measure H2p(SuppT¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∩A)=0 , then T extends to a negative psh current on Ω. We also show that if T is psh or if dd cT extends to a current with locally finite mass on Ω, then the trivial extension T˜ of T by zero across A exists in both cases: A is the zero set of a k−convex function with k≤p−1 or H2(p−1)(SuppT¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∩A)=0 . Our basic tool is the following theorem [El3]: Let A be a closed complete pluripolar subset of an open subset Ω of ℂn and T be a positive current of bidimension (p,p) on Ω\ A. Suppose that T˜ and ddcT˜ exist (resp. T˜ exists and dd cT≤0 on Ω\ A), then there exists a positive (resp. closed positive) current S supported in A such that ddcT˜=ddcT˜+S . Furthermore, we give a generalization of some theorems done by Siu and Ben Messaoud-El Mir and Alessandrini-Bassanelli without requiring anything from dT.