Stable singularities and stable leaves of holomorphic foliations in dimension two

We consider germs of holomorphic foliations with an isolated singularity at the origin $0\in\mathbb C^2$. We introduce a notion of Lstability for the singularity, similar to Lyapunov stability. We prove that $L$-stability is equivalent to the existence of a holomorphic first integral, or the foliation is a real logarithmic foliation. A notion of $L$-stability is also naturally introduced for a leaf of a holomorphic foliation in a complex surface. We prove that the holonomy groups of L-stable leaves are abelian, of a suitable type. This implies the existence of local closed meromorphic $1$-forms defining the foliation, in a neighborhood of compact $L$-stable leaves. Finally, we consider the case of foliations in the complex projective plane. We prove that a foliation on$\mathbb CP^2$ admitting a $L$-stable invariant algebraic curve is the pull-back by some polynomial map of a suitable linear logarithmic foliation.