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 $\mathcal{R}(\alpha )$, 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 $\mathcal{R}(\alpha )=0$ 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 $U^{\ast }$, with $U^{\ast }:={\cup }_{t\in U}{H}_t\subset X$, where $t\times H_t:=p_1^{-1}(t)$. Then $\mathcal{R}(\alpha )$ 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 $\mathcal{R}(\alpha )$ 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 ${\tilde{U}}^{\ast }\subset X$.

11 pages