Preprint A600/2008
Projective limit cycles

Evilson Vieira | Movasati, Hossein

**Keywords: **
holomorphic foliations | holonomy | vanishing cycle

In this article we study projective cycles in $\mathbb{P}^2_\mathbb{R}$. Our inspiring example is the
Jouanolou foliation of odd degree which has a hyperbolic projective limit cycle.
We prove that only odd degree foliations may have projective cycles and
foliations with exactly one real simple singularity have a projective cycle. We also prove that
after a perturbation of a generic Hamiltonian foliation with a projective cycle, we have a projective limit cycle if and only if the perturbation is not Hamiltonian.