Simple dynamics and integrability for singularities of holomorphic foliations in dimension two

Journal of Singularities
volume 14 (2016), 148-171

Received: 20 January 2016. Received in revised form: 5 September 2016.

DOI: 10.5427/jsing.2016.14i

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Abstract:

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.

2000 Mathematical Subject Classification:

37F75, 57R30 (Primary) 32M25, 32S65 (Secondary)

Author(s) information:

Bruno Scárdua
Instituto de Matemática
Universidade Federal do Rio de Janeiro
CP. 68530
Rio de Janeiro
RJ, 21945-970
Brazil
email: scardua@im.ufrj.br