A remark on parabolic projective foliations (Q1301350)

Let \(F\) be a 1-dimensional holomorphic foliation on the complex projective plane \(\mathbb{P}^2\). Then the singular locus \(S:=\text{sing} {\mathcal F}\) of \({\mathcal F}\) is a finite subset of \(\mathbb{P}^2\) and all leaves of \({\mathcal F}\) on \(\mathbb{P}^2 \setminus S\) are noncompact Riemann surfaces. \textit{M. Brunella} [Ann. Inst. Fourier 44, No. 4, 1237-1242 (1994; Zbl 0811.32023)] has shown: if \({\mathcal F}\) has only hyperbolic singularities on \(\mathbb{P}^2\) and the set \({\mathcal P}({\mathcal F}): =\{p\in\mathbb{P}^2 \setminus S:L_p\) is parabolic\} has positive transverse logarithmic capacity then all leaves are parabolic and \({\mathcal F}\) is defined by a differential form \(x dy-\lambda y dx\), \(\lambda\in \mathbb{C}\setminus \mathbb{R}\), in a suitable affine chart. The author proves other theorems of this kind assuming that all leaves are parabolic but replacing the condition of hyperbolic singularities by other conditions. He states many examples.