Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations

David Mathieu

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 4, page 1155-1189
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Mathieu, David. "Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations." Annales de l'institut Fourier 50.4 (2000): 1155-1189. <http://eudml.org/doc/75452>.

@article{Mathieu2000,
abstract = {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.},
author = {Mathieu, David},
journal = {Annales de l'institut Fourier},
keywords = {meromorphic and analytic families of cycles; meromorphic and analytic equivalence relations; analytic structure on a quotient; geometric flattening},
language = {eng},
number = {4},
pages = {1155-1189},
publisher = {Association des Annales de l'Institut Fourier},
title = {Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations},
url = {http://eudml.org/doc/75452},
volume = {50},
year = {2000},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 4
SP - 1155
EP - 1189
AB - 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.
LA - eng
KW - meromorphic and analytic families of cycles; meromorphic and analytic equivalence relations; analytic structure on a quotient; geometric flattening
UR - http://eudml.org/doc/75452
ER -

References

top
  1. [AS71] A. ANDREOTTI, W. STOLL, Analytic and algebraic dependence of meromorphic functions, Lecture Notes 234, Berlin, Springer, 1971. Zbl0222.32013MR52 #11124
  2. [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. Zbl0331.32008MR53 #3347
  3. [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. Zbl0442.32006MR83g:32007
  4. [Ca60] H. CARTAN, Quotients of complex analytic spaces, International Colloquium on Function Theory, Tata Institute (1960), 1-15. Zbl0122.08702MR25 #3199
  5. [Fi76] G. FISCHER, Complex analytic geometry, Lecture Notes 538, Berlin, Springer, 1976. Zbl0343.32002MR55 #3291
  6. [Gr83] H. GRAUERT, Set theoretic complex equivalence relations, Math. Ann., 265 (1983), 137-148. Zbl0504.32007MR85f:32011
  7. [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. Zbl0592.32008MR88b:32023
  8. [Hi73] H. HIRONAKA, La voûte étoilée, Astérisque, 8 (1973), 415-440. Zbl0287.14006MR52 #8494
  9. [HLT73] H. HIRONAKA, M. LEJEUNE-JALABERT, B. TEISSIER, Platificateur local en géométrie analytique et aplatissement local, Astérisque, 8 (1973), 441-463. Zbl0287.14007MR53 #13636
  10. [KK83] L. KAUP, B. KAUP, Holomorphic functions of several variables, Studies in Mathematics 3, Berlin, de Gruyter, 1983. Zbl0528.32001
  11. [Ku64] N. KUHLMANN, Über holomorphe Abbildungen komplexer Räume, Archiv der Math., 15 (1964), 81-90. Zbl0122.08701MR30 #2165
  12. [Ku66] N. KUHLMANN, Bemerkungen über holomorphe Abbildungen komplexer Räume, Wiss. Abh. Arbeitsgemeinschaft Nordrhein-Westfalen 33, Festschr. Gedächtnisfeier K. Weierstrass, 495-522 (1966). Zbl0144.34001MR34 #7826
  13. [Ma74] P. MAZET, Un théorème d'image directe propre, Séminaire Lelong, Lecture Notes, 410 (1974), 107-116; rectificatif Lecture Notes 474 (1975), 180-182. Zbl0305.32014MR51 #10674
  14. [Ma84] P. MAZET, Analytic sets in locally convex spaces, Math. Studies 89, Amsterdam, North Holland, 1984. Zbl0588.46032MR86i:32012
  15. [Mt99] D. MATHIEU, Relations d'équivalence méromorphes et familles analytiques de cycles, Thèse, Nancy, 1999. 
  16. [Pa94] A. PARUSIŃSKI, Subanalytic functions, Trans. Amer. Math. Soc., 344 (1994), 583-595. Zbl0819.32006MR94k:32006
  17. [Re57] R. REMMERT, Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann., 133 (1957), 328-370. Zbl0079.10201MR19,1193d
  18. [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. Zbl0807.32011MR95b:32013

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.