Plurisubharmonic currents and their extension across analytic subsets

Abstract

Some extension problems are considered here for the class of plurisubharmonic currents, i.e. real currents T such that idfiT is positive. In particular we prove the following theorem: "Let Ω be an open subset of C N and Fan analytic subset of Ω. Suppose Tis a negative plurisubharmonic current on Ω -Υ of bidimension (p,p)\ if dim Y<p, then Textende across Υto a plurisubharmonic current on Ω.". For/? = N, this is the classical result of Grauert and Remmert on the extension of plurisubharmonic functions.

Pull-back of currents by holomorphic maps

“…Then Remark 2.3 implies that τ 0 is dsh and hence S |Σ := S − τ 0 is a positive dsh current with support in Σ. By the support theorem [1], this is a current on Σ. One deduces from identity (3.2) that S |Σ , π * (ϕ |A ) = d t T, ϕ |A∩π(Σ) .…”

“…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].…”

A cut-off theorem for plurisubharmonic currents

“…Xi and lim J u n iddL s = J u"iddL; ε~*° 11*11 <s ||*||<s moreover we can apply Lemma 2.1 of [AB2] getting that:…”

“…On the other band, in [AB2] the authors give existence results about the extension of a positive (or negative) plurisubharmonic current Γ across analytic subsets, without requiring anything on dT (for a more detailed discussion see the forthcoming Chapter 3). Therefore normal (and flat) currents don't seem a very appropriate tool for this subject, and we prefer to introduce the less restrictive notions of C-normal and C-flat current (see Definitions 1.17 and 1.1).…”

Regularization of currents and entropy

“…Some of their properties are given in [21,1,2,13,9,14]. Given a compact Kähler manifold (X, ω) of dimension k, we want to define the intersection S ∧ T of a positive closed (1, 1)-current S with a positive pluriharmonic current T of bidegree (p, p), 1 ≤ p ≤ k − 1.…”