J-pluripolar subsets and currents on almost complex manifolds

Abstract

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.

Poincaré–Lelong formula, J -analytic subsets and Lelong numbers of currents on almost complex manifolds

“…Hence, by using an induction on the dimension of A and observe that 1l A j T = 1l A j A j−1 T + 1l A j−1 T , for j ≤ p, we need only to prove that R = 1l A j A j−1 T is positive and closed in Ω. Since A j A j−1 is a smooth almost complex submanifold in Ω A j−1 , again by [2] the current R is positive and closed on Ω A j−1 (notice that R don't carry any mass on A j−1 ). Let's write Ω A j−1 = (Ω A j−2 ) \ (A j−1 A j−2 ).…”

“…Lat ∅ = A −1 ⊂ A 0 ⊂ A 1 ⊂ · · · ⊂ A p = A be a sequence of (Ω, J) as in definition 2 and let T be a positive closed (the case when T is positive and plurisubharmonic is similar) of (Ω, J). Thank's to a result of [2], every almost complex submanifold of (Ω, J) is locally r.c.p and the cut-off of T by a r.c.p subset is also positive and closed. This is the case for 1l A 0 T because A 0 is a smooth almost complex submanifold in Ω.…”

“…Let's write Ω A j−1 = (Ω A j−2 ) \ (A j−1 A j−2 ). Since A j−1 A j−2 is a smooth almost complex submanifold in Ω A j−2 and R has locally finite mass near A j−1 A j−2 then the trivial extension R of R is also positive and closed on Ω A j−2 (see [2]). As the current R is supported by A j A j−1 , we can deduce that R = R. The proof was completed by repeating the above argument and by using the induction hypothesis.…”

Pluri-polarity in almost complex structures

“…That would make the B term negligible. Note that this is asking more than the natural requirement suggested by 'homogeneity' consideration, that would be (|z 2 …”

“…If one wishes to avoid using Jensen measures, one can argue very simply as follows: Fix p ∈ U such that λ 0 ( p) > − r 2 4 , if as above λ = −∞ on A, then for > 0 small enough λ 0 + λ clearly violates the maximum principle that in our case is Sup U = Sup A∪B , unless λ( p) = −∞.…”