Extension of positive current across small subsets

Abstract

The purpose of this paper is to prove an extension result through a closed subset A having small Hausdorff dimension for a positive plurisubharmonic (Psh) current T of dimension p such that ddcT is of locally finite mass. As a consequence we prove that, if there exists a positive (resp. negative) current S such that T = ddcS on Ω\A, then the trivial extension Ŧ of T is a closed positive current in the casewhere A is a Cauchy-Riemann submanifold of class C1 and of dimension CR equal to p − 1.