Pseudomeromorphic currents on subvarieties

Abstract

Let i : X → Y be pure-dimensional reduced subvariety of a smooth manifold Y. We prove that direct images of pseudomeromorphic currents on X are pseudomeromorphic on Y. We also prove a partial converse: if i * τ is pseudomeromorphic and has the standard extension property, then τ is pseudomermorphic on X . ARTICLE HISTORY

Direct images of semi-meromorphic currents

“…, t mr r , which, see, e.g., [1], is the direct image under a modification of a current of the form α∧∂[1/f ], cf., Example 4.18 below. It follows, cf., [5,Lemma 3.2], that (2.4) is the direct image under another modification of a finite sum of elementary currents with at most one residue factor. This proposition is from [10]; for the adaption to nonsmooth X, see [7, Proposition 2.3].…”

The $$\bar{\partial }$$ ∂ ¯ -equation on a non-reduced analytic space

“…For any μ in PM Z there is some μ in PM Z such that π * μ = μ, see [4,Proposition 1.2]. Since μ = τ +∂u with τ, u in W Z , we have that μ = π * τ +∂π * u.…”

“…It is not hard to check that if τ is in PM Z and τ is in PM Z , then τ ⊗ τ is in PM Z ×Z , see, e.g., [4,Lemma 3.3]. If V ⊂ U ⊂ Z and V ⊂ U ⊂ Z , then…”

“…In fact, τ exists on π −1 (X reg ) since π is a biholomorphism there. Moreover, by [4,Proposition 1.2], αh is the direct image of some pseudomeromorphic currentτ on X , and is therefore also the image of the (unique) current τ…”

“…This is done in Sects. [4][5][6][7][8]. The construction of integral operators to provide solutions to∂ in Sect.…”