Finite determinacy of dicritical singularities in ( ℂ 2 , 0 )

Gabriel Calsamiglia-Mendlewicz[1]

  • [1] Universidade Federal Fluminense Departamento de Matemática Aplicada Rua Mário Santos Braga S/N 24020-140 Niterói Rio de Janeiro (Brasil)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 2, page 673-691
  • ISSN: 0373-0956

Abstract

top
For germs of singularities of holomorphic foliations in ( ℂ 2 , 0 ) which are regular after one blowing-up we show that there exists a functional analytic invariant (the transverse structure to the exceptional divisor) and a finite number of numerical parameters that allow us to decide whether two such singularities are analytically equivalent. As a result we prove a formal-analytic rigidity theorem for this kind of singularities.

How to cite

top

Calsamiglia-Mendlewicz, Gabriel. "Finite determinacy of dicritical singularities in $(\mathbb{C}^2,0)$." Annales de l’institut Fourier 57.2 (2007): 673-691. <http://eudml.org/doc/10235>.

@article{Calsamiglia2007,
abstract = {For germs of singularities of holomorphic foliations in $(\mathbb\{C\}^2,0)$ which are regular after one blowing-up we show that there exists a functional analytic invariant (the transverse structure to the exceptional divisor) and a finite number of numerical parameters that allow us to decide whether two such singularities are analytically equivalent. As a result we prove a formal-analytic rigidity theorem for this kind of singularities.},
affiliation = {Universidade Federal Fluminense Departamento de Matemática Aplicada Rua Mário Santos Braga S/N 24020-140 Niterói Rio de Janeiro (Brasil)},
author = {Calsamiglia-Mendlewicz, Gabriel},
journal = {Annales de l’institut Fourier},
keywords = {Dicritical singularities; holomorphic singular foliations; foliations; dicritical singularities, topological classification of germs of foliations; analytic invariants of germs of foliations; finite determinacy of germs of foliations},
language = {eng},
number = {2},
pages = {673-691},
publisher = {Association des Annales de l’institut Fourier},
title = {Finite determinacy of dicritical singularities in $(\mathbb\{C\}^2,0)$},
url = {http://eudml.org/doc/10235},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Calsamiglia-Mendlewicz, Gabriel
TI - Finite determinacy of dicritical singularities in $(\mathbb{C}^2,0)$
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 2
SP - 673
EP - 691
AB - For germs of singularities of holomorphic foliations in $(\mathbb{C}^2,0)$ which are regular after one blowing-up we show that there exists a functional analytic invariant (the transverse structure to the exceptional divisor) and a finite number of numerical parameters that allow us to decide whether two such singularities are analytically equivalent. As a result we prove a formal-analytic rigidity theorem for this kind of singularities.
LA - eng
KW - Dicritical singularities; holomorphic singular foliations; foliations; dicritical singularities, topological classification of germs of foliations; analytic invariants of germs of foliations; finite determinacy of germs of foliations
UR - http://eudml.org/doc/10235
ER -

References

top
  1. G. Calsamiglia, Singularidades Dicríticas e Vizinhanças Folheadas de Superfícies de Riemann, (2005) 
  2. C. Camacho, H. Movasati, P. Sad, Fibred neighbourhoods of curves in surfaces, J. Geom. Anal. 13 (2003), 55-66 Zbl1036.32009MR1967036
  3. C. Camacho, P. Sad, Invariant varieties through holomorphic vector fields, Ann. Math. 115 (1982), 579-595 Zbl0503.32007MR657239
  4. C. Camacho, P. Sad, Pontos singulares de equações diferenciais analíticas, Colóquio Brasileiro de Matemática (1987), IMPA MR953780
  5. H. Cartan, Sur les fonctions de deux variables complexes, Bull. Sci. Math. 54 (1930), 99-116 Zbl56.0981.04
  6. D. Cerveau, J.F. Mattei, Formes intégrables holomorphes singulières, Astérisque 97 (1982) Zbl0545.32006MR704017
  7. M. Klughertz, Feuilletages holomorphes à singularité isolée ayant une infinité de courbes intégrales, (1988) 
  8. J.F. Mattei, Modules de feuilletages holomorphes singuliers. I. Équisingularité, Invent. Math. 103 (1991), 297-325 Zbl0709.32025MR1085109
  9. J.F. Mattei, R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. (4) 13 (1980), 469-523 Zbl0458.32005MR608290
  10. Ortiz-Bobadilla, Rosales-Gonzalez, Voronin, Rigidity theorems for generic holomorphic germs of dicritic foliations and vector fields in ( ℂ 2 , 0 ) , Moscow Math. J. 5 (2005), 171-206 Zbl1091.32012MR2153473
  11. W. Rudin, Function theory in the unit ball of ℂ n , Grundlehren der Mathematischen Wissenschaften 241 (1980), Springer-Verlag, New York-Berlin Zbl0495.32001MR601594
  12. P. Sad, R. Meziani, Singularités nilpotentes et intégrales premières 
  13. M. Suzuki, Sur les intégrales premières de certains feuilletages analytiques complexes. Fonctions de plusieurs variables complexes, III, Lecture Notes in Math., 670 394 (1978), 53-79, Springer, Berlin Zbl0391.32017MR521913
  14. B. Teissier, Singularity theory (Trieste, 1991), (1995), 866-893, World Sci. Publishing, River Edge, NJ Zbl0943.14011MR1378433
  15. O. Zariski, Le problème des modules pour les branches planes, (1975), Centre de Math. de l’École Polytechnique Zbl0317.14004MR414561

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.