Direct images of semi-meromorphic currents

Abstract

We introduce a calculus for the class ASM (X) of direct images of semi-meromorphic currents on a reduded analytic space X, that extends the classical calculus due to Coleff, Herrera and Passare. Our main result is that each element in this class acts as a kind of multiplication on the sheaf PMX of pseudomeromorphic currents on X. We also prove that ASM (X) as well as PMX and certain subsheaves are closed under the action of holomorphic differential operators and interior multiplication by holomorphic vector fields.

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

“…By [10,Theorem 3.7] again τ is locally of the form ξ ∧ ds, where ξ is in W 0, * X and ds = ds 1 ∧ · · · ∧ ds n for some local coordinates s. Hence, τ is a K X -valued section of W 0, * (X ), so τ/π * γ is a section of W 0, * (X ). Now := π * (τ/π * γ ) is a section of W 0, * (X ).…”

“…We end this section with the following result, first part of [10,Theorem 3.7]. We include here a different proof than the one in [10], since we believe the proof here is instructive.…”

“…We include here a different proof than the one in [10], since we believe the proof here is instructive.…”

“…We say that a current a on Z is almost semi-meromorphic if there is a modification π : Z → Z , a holomorphic section f of a line bundle L → Z and a smooth form γ with values in L such that a = π * (γ / f ), cf., [10,Section 4]. If a is almost semi-meromorphic, then it is clearly pseudomeromorphic.…”

Global Koppelman Formulas on (Singular) Projective Varieties

“…where γ is a test form, is elementary. A current τ on X is pseudomeromorphic if locally it is a finite sum of direct images under holomorphic mappings of elementary currents; see, e.g., [10] for a precise definition and basic properties. The pseudomeromorphic currents form a sheaf PM X that is closed under multiplication by E p, * X and the action of∂.…”