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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We prove that every almost complex submanifold of an almost complex manifold is locally J -pluripolar. This generalizes a result of Rosay for J -holomorphic submanifolds. Our second main result is an almost complex version of El Mir's theorem for the extension of positive currents across locally complete pluripolar sets. As a consequence, we extend some results proved by Dabbek-Elkhadhra-El Mir and Dinh-Sibony in the standard complex case. We also obtain a version of the well-known results of Federer and Bassanelli for flat and C-flat currents in the almost complex setting.

In this paper, we first establish a Poincaré-Lelong type formula in the almost complex setting. Then, after introducing the notion of J-analytic subsets, we study the restriction of a closed positive current defined in an almost complex manifold (M, J) on a J-analytic subset. Finally, we prove that the Lelong numbers of a plurisubharmonic current defined on an almost complex manifold are independent of the coordinate systems.