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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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