## Tohoku Mathematical Journal 2002 September SECOND SERIES VOL. 54, NO. 3

 Tohoku Math. J. 54 (2002), 367-392

Title COMPLEX VECTOR FIELDS HAVING ORBITS WITH BOUNDED GEOMETRY

Author Bruno C. A. Scárdua

(Received January 11, 1999, revised February 12, 2002)
Abstract. Germs of holomorphic vector fields at the origin $0\in C^2$ and polynomial vector fields on $C^2$ are studied. Our aim is to classify these vector fields whose orbits have ounded geometry in a certain sense. Namely, we consider the following situations: (i) the volume of orbits is an integrable function, (ii) the orbits have sub-exponential growth, (iii) the total curvature of orbits is finite. In each case we classify these vector fields under some generic hypothesis on singularities. Applications to questions, concerning polynomial vector fields having closed orbits and complete polynomial vector fields, are given. We also give some applications to the classical theory of compact foliations.

2000 Mathematics Subject Classification. Primary 32L30; Secondary 58F18.

Key words and phrases. Singular holomorphic foliation, bounded geometry, holonomy group.