On the Abel-Radon transform of locally residual currents

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 ${\tilde U}^*\subset X$.