On the Abel-Radon transform of locally residual currents

TL;DR

This paper investigates the Abel-Radon transform of locally residual currents on projective varieties, establishing conditions under which the transform's meromorphic extension implies the extension of the currents themselves.

Contribution

It introduces a trace theorem for locally residual currents and characterizes the extension properties of their Abel-Radon transforms on projective varieties.

Findings

The Abel-Radon transform of a locally residual current is meromorphic and holomorphic iff the current is ar{ ext{d}}-closed.

Meromorphic extension of the transform implies the current extends as a locally residual current.

The paper generalizes previous results from projective space to arbitrary projective varieties.

Abstract

First we recall the definition of locally residual currents and their basic properties. We prove in this first section a trace theorem, that we use later. Then we define the Abel-Radon transform of a current ${\cal R}(\alpha)$, on a projective variety $X\subset \P^N$, for a family of $p-$cycles of incidence variety $I\subset T\times X$, for which $p_1:I\to T$ is proper and $p_2:I\to X$ is submersive, and a domain $U\subset T$. Then we show the following theorem, for a family of sections of $X$ with $r-$planes (which was proved for the family of lines of $X=\P^N$ by the author for $p=1$, for ${\cal R}(\alpha)=0$ and $p-$planes for any $q>0$, and by Henkin and Passare for $p-$planes in $\P^N$ and integration currents $\alpha=\omega\wedge[Y]$, with a meromorphic $q-$form $\omega$, and projective convexity on $\tilde U$): Let $\alpha$ be a locally residual current of bidegree $(q+p,p)$ on $U^*$, with $U^*:=\cup_{t\in U}{H_t}\subset X$, where ${t}\times H_t:=p_1^{-1}(t)$. Then ${\cal R}(\alpha)$ is a meromorphic $q-$form on $U$, holomorphic iff $\alpha$ is $\bar{\partial}-$closed. Let us assume that $\alpha$ is $\bar\partial-$closed, and $q>0$. If ${\cal R}(\alpha)$ extends meromorphically (resp. holomorphically) to a greater domain $\tilde U$, then $\alpha$ extends in a unique way as a locally residual current (resp. $\bar\partial-$closed) to the greater domain ${\tilde U}^*\subset X$.