Pseudomeromorphic currents on subvarieties

TL;DR

This paper studies pseudomeromorphic currents on subvarieties and proves that their direct images preserve pseudomeromorphicity, with a partial converse under certain extension properties.

Contribution

It establishes the behavior of pseudomeromorphic currents under direct image operations and provides conditions for their converse on subvarieties.

Findings

Direct images of pseudomeromorphic currents are pseudomeromorphic on the ambient manifold.

A partial converse shows pseudomeromorphicity can be inferred under the standard extension property.

Results extend the understanding of currents in complex geometry.

Abstract

Let $i\colon X \to Y$ be pure-dimensional reduced subvariety of a smooth manifold $Y$. We prove that the direct image of pseudomeromorphic currents on $X$ are pseudomeromorphic on $Y$. We also prove a partial converse: if $i_*\tau$ is pseudomeromorphic and has the standard extension property, then $\tau$ is pseudomermorphic on $X$.