Sur la transformation d'Abel-Radon des courants localement résiduels
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 1, p. 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 (α) of a locally residual current α 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 N , U * := tU H t be the corresponding linearly p-concave domain of N , and α be a locally residual current of bidegree (N,p). Suppose that the meromorphic n-form (α) extends meromorphically to a greater domain U ˜ of G(p,N). If α is of type ω[T], with T an analytic subvariety of pure codimension p in U * , and ω a meromorphic (resp. regular) q-form (q>0) on T, then α extends in a unique way as a locally residual current to the domain U ˜ * := tU ˜ H t . In particular, if (α)=0, then α extends as a ¯-closed residual current on 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.

Classification:  32C30,  44A12
@article{ASNSP_2005_5_4_1_27_0,
     author = {Fabre, Bruno},
     title = {Sur la transformation d'Abel-Radon des courants localement r\'esiduels},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {5e s{\'e}rie, 4},
     number = {1},
     year = {2005},
     pages = {27-57},
     zbl = {1170.32305},
     mrnumber = {2165402},
     language = {fr},
     url = {http://www.numdam.org/item/ASNSP_2005_5_4_1_27_0}
}
Fabre, Bruno. Sur la transformation d'Abel-Radon des courants localement résiduels. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 1, pp. 27-57. http://www.numdam.org/item/ASNSP_2005_5_4_1_27_0/

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