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

Fabre, Bruno

Fabre, Bruno

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 1, p. 27-57

Abstract

After recalling the definitions of the Abel-Radon transformation of currents and of locally residual currents, we show that the Abel-Radon transform $ℛ (α)$ 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 $¶^{N}$ , $U^{*} : = \cup_{t \in U} H_{t}$ 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 $ℛ (α)$ extends meromorphically to a greater domain $\tilde{U}$ of $G (p, N)$ . If $α$ is of type $ω \land [T]$ , 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 ${\tilde{U}}^{*} : = \cup_{t \in \tilde{U}} H_{t}$ . In particular, if $ℛ (α) = 0$ , then $α$ extends as a $\bar{\partial}$ -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 $p = 1$ .

MR 2165402 | Zbl 1170.32305 | 1 citation in Numdam

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/

References

[1] J.-E Björk, Residues and $𝒟$ -modules, In : “The legacy of Niels Henrik Abel”, The Abel Bicentennial, Oslo, Springer, 2002, 605-651. | MR 2077588 | Zbl 1069.32001

[2] N. Coleff and M. Hererra, “Les courants résiduels associés à une forme méromorphe”, Springer Lect. Notes 633, 1978. | Zbl 0371.32007

[3] A. Dickenstein, M. Hererra and C. Sessa, On the global lifting of meromorphic forms, Manuscripta Math. 47 (1984), 31-45. | MR 744312 | Zbl 0565.32012

[4] A. Dickenstein and C. Sessa, Canonical representative in moderate cohomology, Invent. Math. 80 (1985), 417-434. | MR 791667 | Zbl 0556.32005

[5] P. Dingoyan, Un phénomène de Hartogs dans les variétés projectives, Math. Z. 232 (1999), 217-240. | MR 1718681 | Zbl 0941.32011

[6] B. Fabre, 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] P. Griffiths, Variations on a theorem of Abel, Invent. math. 35 (1976), 321-390. | MR 435074 | Zbl 0339.14003

[8] S. G. Gindikin and G. M. Henkin, Integral geometry for $\bar{\partial}$ -cohomology in $q$ -linear concave domains in $C P^{n}$ , Functional Anal. Appl. 12 (1978), 247-261. | MR 515626 | Zbl 0423.32013

[9] G. Henkin, The Abel-Radon transform and several complex variables, Ann. of Math. Studies 137 (1995), 223-275. | MR 1369141 | Zbl 0848.32012

[10] G. Henkin and M. Passare, Abelian differentials on singular varities and variations on a theorem of Lie-Griffiths, Invent. Math. 135 (1999), 297-328. | MR 1666771 | Zbl 0932.32012

[11] G. Henkin, 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] M. Hererra and D. Lieberman, Residues and principal values on complex spaces, Math. Ann. 194 (1971), 259-294. | MR 296352 | Zbl 0224.32012

[13] H. Kneser, Einfacher Beweis eines Satzes über rationale Funktionen zweier Veränderlichen, Hamburg Univ. Math. Sem. Abhandl. 9 (1933), 195-196. | JFM 59.0357.02

[14] M. Passare, Residues, currents, and their relations to ideals of meromorphic functions, Math. Scand. 62 (1988), 75-152. | MR 961584 | Zbl 0633.32005

[15] R. Remmert and K. Stein, Über die wesentlichen Singularitäten analytischer Mengen, Math. Ann. 126 (1953), 263-306. | MR 60033 | Zbl 0051.06303

[16] W. Rothstein, Ein neuer Beweis des Hartogsschen Hauptsatzes und seine Ausdehnung auf meromorphe Funktionen, Math. Z. 53 (1950), 84-95. | MR 37365 | Zbl 0037.18301

[17] L. Schwartz, “Théorie des distributions”, 2nd ed., Hermann, Paris, 1966. | Zbl 0149.09501

[18] J. A. Wood, A simple criterion for local hypersurfaces to be algebraic, Duke Math. J. 51 (1984), 235-237. | MR 744296 | Zbl 0584.14021