Abstract

After recalling the definitions of the Abel-Radon transformation of currents and of locally residual currents, we show that the Abel-Radon transform ${\mathcal{R}}$ of a locally residual current $\alpha $ remains locally residual. Then a theorem of P. Griffiths, G. Henkin and M. Passare can be formulated as follows : Let $U$ be a domain of the grassmannian variety $G$ of complex $p$-planes in $¶^N$, $U^*\!:=\!\cup _{t\in U}{H_t}$ be the corresponding linearly $p$-concave domain of $¶^N$, and $\alpha $ be a locally residual current of bidegree $$. Suppose that the meromorphic $n$-form ${\mathcal{R}}$ extends meromorphically to a greater domain $\tilde{U}$ of $G$. If $\alpha $ is of type $\omega \wedge [T]$, with $T$ an analytic subvariety of pure codimension $p$ in $U^*$, and $\omega $ a meromorphic $q$-form on $T$, then $\alpha $ extends in a unique way as a locally residual current to the domain ${\tilde{U}}^*\!:=\!\cup _{t\in {\tilde{U}}}{H_t}$. In particular, if ${\mathcal{R}}\!=\!0$, then $\alpha $ extends as a $\overline{\partial }$-closed residual current on $¶^N$. We show in this note that this theorem remains valid for an arbitrary residual current of bidegree $$, in the particular case where $p=1$