Let A be a closed subset of an open subset of C n and T be a negative current on A of bidimension (p, p). Assume that T is psh and A is complete pluripolar such that the Hausdorff measure H 2p (SuppT ∩ A) = 0, then T extends to a negative psh current on . We also show that if T is psh or if dd c T extends to a current with locally finite mass on , 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 H 2(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 of C n and T be a positive current of bidimension (p, p) on A. Suppose that T and dd c T exist (resp. T exists and dd c T ≤ 0 on A), then there exists a positive (resp. closed positive) current S supported in A such that dd c T = dd c T + 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 .

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