no. 4
Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations
Annales de l'Institut Fourier, Tome 50 (2000) no. 4, pp. 1155-1189.

Nous étudions des familles analytiques de cycles non-compacts, et prouvons qu’il existe un espace analytique de dimension finie, qui fournit une reparamétrisation universelle d’une telle famille, sous certaines conditions de régularité. Nous démontrons ensuite un résultat analogue pour les familles méromorphes de cycles non-compacts. C’est une nouvelle approche des résultats de Grauert sur les relations d’équivalence méromorphes.

We study analytic families of non-compact cycles, and prove there exists an analytic space of finite dimension, which gives a universal reparametrization of such a family, under some assumptions of regularity. Then we prove an analogous statement for meromorphic families of non-compact cycles. That is a new approach to Grauert’s results about meromorphic equivalence relations.

@article{AIF_2000__50_4_1155_0,
     author = {Mathieu, David},
     title = {Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations},
     journal = {Annales de l'Institut Fourier},
     pages = {1155--1189},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {4},
     year = {2000},
     doi = {10.5802/aif.1788},
     mrnumber = {2002c:32011},
     zbl = {0966.32003},
     language = {en},
     url = {http://www.numdam.org./articles/10.5802/aif.1788/}
}
TY  - JOUR
AU  - Mathieu, David
TI  - Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations
JO  - Annales de l'Institut Fourier
PY  - 2000
SP  - 1155
EP  - 1189
VL  - 50
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org./articles/10.5802/aif.1788/
DO  - 10.5802/aif.1788
LA  - en
ID  - AIF_2000__50_4_1155_0
ER  - 
%0 Journal Article
%A Mathieu, David
%T Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations
%J Annales de l'Institut Fourier
%D 2000
%P 1155-1189
%V 50
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org./articles/10.5802/aif.1788/
%R 10.5802/aif.1788
%G en
%F AIF_2000__50_4_1155_0
Mathieu, David. Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations. Annales de l'Institut Fourier, Tome 50 (2000) no. 4, pp. 1155-1189. doi : 10.5802/aif.1788. http://www.numdam.org./articles/10.5802/aif.1788/

[AS71] A. Andreotti, W. Stoll, Analytic and algebraic dependence of meromorphic functions, Lecture Notes 234, Berlin, Springer, 1971. | MR | Zbl

[Ba75] D. Barlet, Espace analytique réduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie, Séminaire Norguet, Lecture Notes 482 (1975), 1-158. | MR | Zbl

[Ba78] D. Barlet, Majoration du volume des fibres génériques et forme géométrique du théorème d'aplatissement, Séminaire Lelong-Skoda 1978-79, Lecture Notes 822 (1980), 1-17. | MR | Zbl

[Ca60] H. Cartan, Quotients of complex analytic spaces, International Colloquium on Function Theory, Tata Institute (1960), 1-15. | MR | Zbl

[Fi76] G. Fischer, Complex analytic geometry, Lecture Notes 538, Berlin, Springer, 1976. | MR | Zbl

[Gr83] H. Grauert, Set theoretic complex equivalence relations, Math. Ann., 265 (1983), 137-148. | MR | Zbl

[Gr86] H. Grauert, On meromorphic equivalence relations, in Contributions to several complex variables, Proc. Conf. Complex Analysis, Notre Dame, Indiana 1984, Aspects Math., E9 (1986), 115-147. | MR | Zbl

[Hi73] H. Hironaka, La voûte étoilée, Astérisque, 8 (1973), 415-440. | Numdam | MR | Zbl

[HLT73] H. Hironaka, M. Lejeune-Jalabert, B. Teissier, Platificateur local en géométrie analytique et aplatissement local, Astérisque, 8 (1973), 441-463. | Numdam | MR | Zbl

[KK83] L. Kaup, B. Kaup, Holomorphic functions of several variables, Studies in Mathematics 3, Berlin, de Gruyter, 1983. | Zbl

[Ku64] N. Kuhlmann, Über holomorphe Abbildungen komplexer Räume, Archiv der Math., 15 (1964), 81-90. | MR | Zbl

[Ku66] N. Kuhlmann, Bemerkungen über holomorphe Abbildungen komplexer Räume, Wiss. Abh. Arbeitsgemeinschaft Nordrhein-Westfalen 33, Festschr. Gedächtnisfeier K. Weierstrass, 495-522 (1966). | MR | Zbl

[Ma74] P. Mazet, Un théorème d'image directe propre, Séminaire Lelong, Lecture Notes, 410 (1974), 107-116; rectificatif Lecture Notes 474 (1975), 180-182. | MR | Zbl

[Ma84] P. Mazet, Analytic sets in locally convex spaces, Math. Studies 89, Amsterdam, North Holland, 1984. | MR | Zbl

[Mt99] D. Mathieu, Relations d'équivalence méromorphes et familles analytiques de cycles, Thèse, Nancy, 1999.

[Pa94] A. Parusiński, Subanalytic functions, Trans. Amer. Math. Soc., 344 (1994), 583-595. | MR | Zbl

[Re57] R. Remmert, Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann., 133 (1957), 328-370. | EuDML | MR | Zbl

[Si93] B. Siebert, Fibre cycles of holomorphic maps, I. Local flattening, Math. Ann., 296 (1993), 269-283; Fibre cycle space and canonical flattening, II, Math. Ann. 300 (1994), 243-271. | EuDML | MR | Zbl

Cité par Sources :