Monodromy and topological classification of germs of holomorphic foliations

David Marín; Jean-François Mattei

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 3, page 405-445
  • ISSN: 0012-9593

Abstract

top
We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective holonomy representations whose topological invariance was conjectured in the eighties by Cerveau and Sad and is proved here under mild hypotheses.

How to cite

top

Marín, David, and Mattei, Jean-François. "Monodromy and topological classification of germs of holomorphic foliations." Annales scientifiques de l'École Normale Supérieure 45.3 (2012): 405-445. <http://eudml.org/doc/272135>.

@article{Marín2012,
abstract = {We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective holonomy representations whose topological invariance was conjectured in the eighties by Cerveau and Sad and is proved here under mild hypotheses.},
author = {Marín, David, Mattei, Jean-François},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {differential equations; holomorphic foliations; singularities; monodromy; holonomy},
language = {eng},
number = {3},
pages = {405-445},
publisher = {Société mathématique de France},
title = {Monodromy and topological classification of germs of holomorphic foliations},
url = {http://eudml.org/doc/272135},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Marín, David
AU - Mattei, Jean-François
TI - Monodromy and topological classification of germs of holomorphic foliations
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 3
SP - 405
EP - 445
AB - We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective holonomy representations whose topological invariance was conjectured in the eighties by Cerveau and Sad and is proved here under mild hypotheses.
LA - eng
KW - differential equations; holomorphic foliations; singularities; monodromy; holonomy
UR - http://eudml.org/doc/272135
ER -

References

top
  1. [1] C. Camacho, A. Lins Neto & P. Sad, Topological invariants and equidesingularization for holomorphic vector fields, J. Differential Geom.20 (1984), 143–174. Zbl0576.32020
  2. [2] C. Camacho & P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math.115 (1982), 579–595. Zbl0503.32007MR657239
  3. [3] D. Cerveau & P. Sad, Problèmes de modules pour les formes différentielles singulières dans le plan complexe, Comment. Math. Helv.61 (1986), 222–253. Zbl0604.58004
  4. [4] R. Douady & A. Douady, Algèbre et théories galoisiennes, 2nd éd., Cassini, 2005. Zbl1076.12004
  5. [5] W. Heil, Normalizers of incompressible surfaces in 3 -manifolds, Glas. Mat. Ser. III 16 (36) (1981), 145–150. Zbl0475.57001MR634302
  6. [6] L. Le Floch, Rigidité générique des feuilletages singuliers, Ann. Sci. École Norm. Sup.31 (1998), 765–785. Zbl0934.32023MR1664226
  7. [7] F. Loray, Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux, preprint http://hal.inria.fr/docs/00/05/37/08/PDF/LeconsLink.pdf, 2005. 
  8. [8] D. Marín, Moduli spaces of germs of holomorphic foliations in the plane, Comment. Math. Helv.78 (2003), 518–539. Zbl1054.32018MR1998392
  9. [9] D. Marín & J.-F. Mattei, Incompressibilité des feuilles de germes de feuilletages holomorphes singuliers, Ann. Sci. Éc. Norm. Supér. 41 (2008), 855–903. Zbl1207.32028MR2504107
  10. [10] D. Marín & J.-F. Mattei, Mapping class group of a plane curve germ, Topology Appl.158 (2011), 1271–1295. Zbl1273.32034MR2806361
  11. [11] D. Marín & J.-F. Mattei, Monodromie et classification topologique de germes de feuilletages holomorphes, preprint arXiv:1004.1552. Zbl1308.32036
  12. [12] J.-F. Mattei & R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup.13 (1980), 469–523. Zbl0458.32005MR608290
  13. [13] J. Milnor, Singular points of complex hypersurfaces, Annals of Math. Studies, No. 61, Princeton Univ. Press, 1968. Zbl0184.48405MR239612
  14. [14] L. Ortiz-Bobadilla, E. Rosales-González & S. M. Voronin, Extended holonomy and topological invariance of vanishing holonomy group, J. Dyn. Control Syst.14 (2008), 299–358. Zbl1203.32012MR2425303
  15. [15] L. Ortiz-Bobadilla, E. Rosales-Gonzalez & S. M. Voronin, On Camacho-Sad’s theorem about the existence of a separatrix, Internat. J. Math.21 (2010), 1413–1420. Zbl1207.32029MR2747734
  16. [16] J. C. Rebelo, On transverse rigidity for singular foliations in ( ℂ 2 , 0 ) , Ergodic Theory Dynam. Systems31 (2011), 935–950. Zbl1232.37030MR2794955
  17. [17] R. Rosas, Constructing equivalences with some extensions to the divisor and topological invariance of projective holonomy, preprint, 2012. 
  18. [18] A. Seidenberg, Reduction of singularities of the differential equation A d y = B d x , Amer. J. Math.90 (1968), 248–269. Zbl0159.33303MR220710

NotesEmbed ?

top

You must be logged in to post comments.