Complex projective foliations having sub-exponential growth (Q1605092)

Let \(X\) be a polynomial vector field on the complex plane, and denote by \({\mathcal F}\) the corresponding holomorphic foliation on the complex projective plane. Assume that \({\mathcal F}\) has hyperbolic singularities and for some Riemannian metric (hermitian along the leaves), these leaves have sub-exponential growth. The author shows that \({\mathcal F}\) arose from a linear and hyperbolic polynomial vector field, on some suitable affine chart. In particular, the limit set of the foliation is a union of singularities and algebraic leaves. The author uses the classical ideas of J. Plante on the growth of the leaves for real foliations. This is a nice application, since this technique is almost unexplored for holomorphic foliations.