Singularities of complex vector fields having many closed orbits

Bruno Scárdua

Journal of Singularities
volume 8 (2014), 69-82

Received: 7 June 2014. Received in revised form: 5 September 2014.

DOI: 10.5427/jsing.2014.8f

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Abstract:

A well-known result of Mattei and Moussu states that a germ of a holomorphic vector field at the origin in C^2 admits a holomorphic first integral if, and only if, the orbits are closed off the origin and only finitely many of these accumulate (only) at the origin. In this paper we investigate possible versions of such a result in terms of the measure of the set of closed orbits. We prove that if the set of closed leaves is a positive, i.e., a non-zero measure subset and the set of leaves accumulating only at the origin is a zero measure subset, then either there is a holomorphic first integral or the germ is formally linearizable as a suitable non-resonant singularity. The result is sharp as we show through some examples.

Keywords:

Singular holomorphic foliation, holonomy group, closed orbit

Mathematical Subject Classification:

Primary 37F75, 57R30; Secondary 32M25, 32S65

Author(s) information:

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