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The Abel-Radon transformation of locally residual currents. (Sur la transformation d’Abel-Radon des courants localement résiduels.) (French. English summary) Zbl 1170.32305
Summary: After recalling the definitions of the Abel-Radon transformation of currents and of locally residual currents, we show that the Abel-Radon transform $${\mathcal R} (\alpha)$$ of a locally residual current a remains locally residual. Then a theorem of P. Griffiths [Invent. Math. 35, 321–390 (1976; Zbl 0339.14003)], G. Henkin [Ann. Math. Stud. 137, 223–275 (1995; Zbl 0848.32012)], M. Henkin and M. Passare [Invent. Math. 135, No. 2, 297–328 (1999; Zbl 0932.32012)], can be formulated as follows: Let $$U$$ be a domain of the Grassmannian variety $$G(p,N)$$ of complex $$p$$-planes in $$\mathbb{P}^N$$, $$U^*:=\cup_{t\in U}H_t$$ be the corresponding linearly $$p$$-concave domain of $$\mathbb{P}^N$$, and $$\alpha$$ be a locally residual current of bidegree $$(N,p)$$. Suppose that the meromorphic $$n$$-form $${\mathcal R} (\alpha)$$ extends meromorphically to a greater domain $$\widetilde U$$ of $$G(p,N)$$. If $$\alpha$$ is of type $$\omega\wedge[T]$$, 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 $$\widetilde U^*:= \cup_{t\in\widetilde U}H_t$$. In particular, if $${\mathcal R} (\alpha)=0$$, then $$\alpha$$ extends as a $$\overline\partial$$-closed residual current on $$\mathbb{P}^N$$. We show in this note that this theorem remains valid for an arbitrary residual current of bidegree $$(N,p)$$, in the particular case where $$p=1$$.

##### MSC:
 32C30 Integration on analytic sets and spaces, currents
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##### References:
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