Sur la transformation d'Abel-Radon des courants localement résiduels

Fabre, Bruno

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 27-57.

After recalling the definitions of the Abel-Radon transformation of currents and of locally residual currents, we show that the Abel-Radon transform $ℛ\left(\alpha \right)$ of a locally residual current $\alpha$ remains locally residual. Then a theorem of P. Griffiths, G. Henkin and M. Passare (cf. [7], [9] and [10]) can be formulated as follows  : Let $U$ be a domain of the grassmannian variety $G(p,N)$ of complex $p$-planes in ${\P }^N$, $U^{*}:={\cup }_{t\in U}{H}_t$ be the corresponding linearly $p$-concave domain of ${\P }^N$, and $\alpha$ be a locally residual current of bidegree $(N,p)$. Suppose that the meromorphic $n$-form $ℛ\left(\alpha \right)$ extends meromorphically to a greater domain $\tilde{U}$ of $G(p,N)$. If $\alpha$ is of type $\omega \wedge \left[T\right]$, with $T$ an analytic subvariety of pure codimension $p$ in $U^{*}$, and $\omega$ a meromorphic (resp. regular) $q$-form ($q>0$) 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 $ℛ\left(\alpha \right)=0$, then $\alpha$ extends as a $\overline{\partial }$-closed residual current on ${\P }^N$. We show in this note that this theorem remains valid for an arbitrary residual current of bidegree $(N,p)$, in the particular case where $p=1$.

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