After recalling the definitions of the Abel-Radon transformation of currents and of locally residual currents, we show that the Abel-Radon transform of a locally residual current 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 be a domain of the grassmannian variety of complex -planes in , be the corresponding linearly -concave domain of , and be a locally residual current of bidegree . Suppose that the meromorphic -form extends meromorphically to a greater domain of . If is of type , with an analytic subvariety of pure codimension in , and a meromorphic (resp. regular) -form () on , then extends in a unique way as a locally residual current to the domain . In particular, if , then extends as a -closed residual current on . We show in this note that this theorem remains valid for an arbitrary residual current of bidegree , in the particular case where .
@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}, pages = {27--57}, publisher = {Scuola Normale Superiore, Pisa}, volume = {5e s{\'e}rie, 4}, number = {1}, year = {2005}, zbl = {1170.32305}, mrnumber = {2165402}, language = {fr}, url = {archive.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, Série 5, Tome 4 (2005) no. 1, pp. 27-57. http://archive.numdam.org/item/ASNSP_2005_5_4_1_27_0/
[1] Residues and -modules, In : “The legacy of Niels Henrik Abel”, The Abel Bicentennial, Oslo, Springer, 2002, 605-651. | MR 2077588 | Zbl 1069.32001
,[2] “Les courants résiduels associés à une forme méromorphe”, Springer Lect. Notes 633, 1978. | Zbl 0371.32007
and ,[3] On the global lifting of meromorphic forms, Manuscripta Math. 47 (1984), 31-45. | MR 744312 | Zbl 0565.32012
, and ,[4] Canonical representative in moderate cohomology, Invent. Math. 80 (1985), 417-434. | MR 791667 | Zbl 0556.32005
and ,[5] Un phénomène de Hartogs dans les variétés projectives, Math. Z. 232 (1999), 217-240. | MR 1718681 | Zbl 0941.32011
,[6] On a problem of Griffiths : inversion of Abel's theorem for families of zero-cycles, Ark. Mat. 41 (2003), 61-84. | MR 1971940 | Zbl 1035.14002
,[7] Variations on a theorem of Abel, Invent. math. 35 (1976), 321-390. | MR 435074 | Zbl 0339.14003
,[8] Integral geometry for -cohomology in -linear concave domains in , Functional Anal. Appl. 12 (1978), 247-261. | MR 515626 | Zbl 0423.32013
and ,[9] The Abel-Radon transform and several complex variables, Ann. of Math. Studies 137 (1995), 223-275. | MR 1369141 | Zbl 0848.32012
,[10] Abelian differentials on singular varities and variations on a theorem of Lie-Griffiths, Invent. Math. 135 (1999), 297-328. | MR 1666771 | Zbl 0932.32012
and ,[11] Abel-Radon transform and applications, In : “The legacy of Niels Henrik Abel”, Springer, The Abel Bicentennial, Oslo, 2002, 477-494. | MR 2077585 | Zbl 1075.44002
,[12] Residues and principal values on complex spaces, Math. Ann. 194 (1971), 259-294. | MR 296352 | Zbl 0224.32012
and ,[13] Einfacher Beweis eines Satzes über rationale Funktionen zweier Veränderlichen, Hamburg Univ. Math. Sem. Abhandl. 9 (1933), 195-196. | JFM 59.0357.02
,[14] Residues, currents, and their relations to ideals of meromorphic functions, Math. Scand. 62 (1988), 75-152. | MR 961584 | Zbl 0633.32005
,[15] Über die wesentlichen Singularitäten analytischer Mengen, Math. Ann. 126 (1953), 263-306. | MR 60033 | Zbl 0051.06303
and ,[16] Ein neuer Beweis des Hartogsschen Hauptsatzes und seine Ausdehnung auf meromorphe Funktionen, Math. Z. 53 (1950), 84-95. | MR 37365 | Zbl 0037.18301
,[17] “Théorie des distributions”, 2nd ed., Hermann, Paris, 1966. | Zbl 0149.09501
,[18] A simple criterion for local hypersurfaces to be algebraic, Duke Math. J. 51 (1984), 235-237. | MR 744296 | Zbl 0584.14021
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