Simple dynamics and integrability for singularities of holomorphic foliations in dimension two
Journal of Singularities, Tome 14 (2016), pp. 148-171
Voir la notice de l'article provenant de la source Journal of Singularities website
In this paper we study the dynamics of a holomorphic vector field near a singular point in dimension two. We consider those for which the set of separatrices is finite and the orbits are closed off this analytic set. We assume that none of the singularities arising in the reduction of the foliation has a zero eigenvalue. Under these hypotheses we prove that one of the following cases occurs: (i) there is a holomorphic first integral, (ii) the induced foliation is a pull-back of a hyperbolic linear singularity, (iii) there is a formal Liouvillian first integral. For a germ with closed leaves off the set of separatrices we prove that the existence of a holomorphic first integral is equivalent to the existence of some closed leaf arbitrarily close to the singularity. For this we do not need to assume any non-degeneracy hypothesis on the reduction of singularities. We also study some examples illustrating our results and we prove a characterization of pull-backs of hyperbolic singularities in terms of the dynamics of the leaves off the set of separatrices.
@article{10_5427_jsing_2016_14i,
author = {Bruno Sc\'ardua},
title = {Simple dynamics and integrability for singularities of holomorphic foliations in dimension two},
journal = {Journal of Singularities},
pages = {148--171},
publisher = {mathdoc},
volume = {14},
year = {2016},
doi = {10.5427/jsing.2016.14i},
url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2016.14i/}
}
TY - JOUR AU - Bruno Scárdua TI - Simple dynamics and integrability for singularities of holomorphic foliations in dimension two JO - Journal of Singularities PY - 2016 SP - 148 EP - 171 VL - 14 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.5427/jsing.2016.14i/ DO - 10.5427/jsing.2016.14i ID - 10_5427_jsing_2016_14i ER -
%0 Journal Article %A Bruno Scárdua %T Simple dynamics and integrability for singularities of holomorphic foliations in dimension two %J Journal of Singularities %D 2016 %P 148-171 %V 14 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.5427/jsing.2016.14i/ %R 10.5427/jsing.2016.14i %F 10_5427_jsing_2016_14i
Bruno Scárdua. Simple dynamics and integrability for singularities of holomorphic foliations in dimension two. Journal of Singularities, Tome 14 (2016), pp. 148-171. doi: 10.5427/jsing.2016.14i
Cité par Sources :