Extension of plurisubharmonic currents

Abstract

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 .

Pull-back of currents by holomorphic maps

“…For S = 0 it is proved by Dabbek-Elkhadhara-El Mir in [5], see also Remark 2.3, and is due to Alessandrini-Bassanelli [1,3] when F is an analytic set and dd c T has bounded mass. Under the extra assumption that dT is of order zero, the result was proved by the second author in [20].…”

J-pluripolar subsets and currents on almost complex manifolds

“…The purpose of the second section is to present a generalization of El Mir's theorem [4,8] on the extension of positive currents across a complete pluripolar subset, in the almost complex setting. In the case of closed currents especially, El Mir's theorem had found quite a number of substantial applications for the development of complex analysis during the last 25 years (see [4] for more comments).…”

“…In the case of closed currents especially, El Mir's theorem had found quite a number of substantial applications for the development of complex analysis during the last 25 years (see [4] for more comments). Let (M, J ) be an almost complex manifold, A a closed subset of M and T a current of order zero on M A.…”

“…(the argument consists of considering a suitable convergent series ε m u m and extends without difficulty to the almost complex case). Now, we are ready to state the almost complex version of El Mir's theorem: In the case when J is integrable, this reduces to the results of Dabbek et al [4], see also [8] for the closed case. We mention here that the main result of [4] is to prove Theorem 1 in the complex situation without requiring anything on the mass of the current dT .…”

Prolongement d’un courant positif quasi-plurisurharmonique

“…Le cas d'un courant positif fermé est démontré par El Mir-Feki [8], le cas d'un courant négatif plurisousharmonique (S = 0) est démontré par Dabbek, Elkhadhra et El Mir [4], alors que le cas d'un courant positif plurisousharmonique (psh) où son dd c est supposé de masse localement finie est démontré par Dabbek [2] (en choisissant S = dd c T ). …”