Nodal separators of holomorphic foliations
Annales de l'Institut Fourier, Volume 68 (2018) no. 2, p. 511-539
We study a special kind of local invariant sets of singular holomorphic foliations called nodal separators. We define notions of equisingularity and topological equivalence for nodal separators as intrinsic objects and, in analogy with the celebrated theorem of Zariski for analytic curves, we prove the equivalence of these notions. We give some applications in the study of topological equivalences of holomorphic foliations. In particular, we show that the nodal singularities and its eigenvalues in the resolution of a generalized curve are topological invariants.
Nous étudions un type particulier d’ensembles invariants locaux de feuilletages holomorphes singuliers appelés séparateurs nodaux. Nous définissons des notions d’équisingularité et d’équivalence topologique pour les séparateurs nodaux comme des objets intrinsèques et, par analogie avec le célèbre théorème de Zariski pour les courbes analytiques, nous prouvons l’équivalence de ces notions. Nous donnons quelques applications à l’étude des équivalences topologiques de feuilletages holomorphes. En particulier, nous montrons que les singularités nodales et ses valeurs propres dans la résolution d’une courbe généralisée sont des invariants topologiques.
Received : 2016-12-06
Revised : 2017-03-15
Accepted : 2017-04-28
Published online : 2018-04-18
DOI : https://doi.org/10.5802/aif.3168
Classification:  37F75,  34M35,  32S15
Keywords: Holomorphic foliation, topological equivalence, equisingularity
@article{AIF_2018__68_2_511_0,
     author = {Rosas, Rudy},
     title = {Nodal separators of holomorphic foliations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {2},
     year = {2018},
     pages = {511-539},
     doi = {10.5802/aif.3168},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_2_511_0}
}
Nodal separators of holomorphic foliations. Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 511-539. doi : 10.5802/aif.3168. https://aif.centre-mersenne.org/item/AIF_2018__68_2_511_0/

[1] Camacho, César; Lins Neto, Alcides; Sad, Paulo Topological invariants and equidesingularization for holomorphic vector fields, J. Differ. Geom., Tome 20 (1984) no. 1, pp. 143-174 | Article | MR MR772129 (86d:58080) | Zbl 0576.32020

[2] Camacho, César; Rosas, Rudy Invariant sets near singularities of holomorphic foliations, Ergodic Theory Dyn. Syst., Tome 36 (2016) no. 8, pp. 2408-2418 | Article | MR 3570017 | Zbl 1373.37115

[3] Marín, David; Mattei, Jean-François Monodromy and topological classification of germs of holomorphic foliations, Ann. Sci. Éc. Norm. Supér., Tome 45 (2012) no. 3, pp. 405-445 | Article | MR 3014482 | Zbl 1308.32036

[4] Rosas, Rudy Constructing equivalences with some extensions to the divisor and topological invariance of projective holonomy, Comment. Math. Helv., Tome 89 (2014) no. 3, pp. 631-670 | Article | MR 3260845 | Zbl 06361416

[5] Zariski, Oscar On the Topology of Algebroid Singularities, Am. J. Math., Tome 54 (1932) no. 3, pp. 453-465 | Article | MR 1507926 | Zbl 0004.36902