Consider a vector field $v$ in $ℂ^2$ given by $\dot x=P(x, y)$, $\dot y=Q(x, y)$, where $P$ and $Q$ are polynomials of degree $n$. The geometric properties of $v$ for generic $P$, $Q$ are very different from those of a generic polynomial vector field in $\mathbb{R}^2$. I shall give a very brief overview of these differences, and discuss some recent result.
1. (in collaboration with V. Ramirez)
For a generic quadratic vector field, consider the spectra of its finite zeros, and the Camacho-Sad indices of its singularities at the infinite line. Well-known index theorems imply four relations on these numbers, while a simple dimension count shows that there should be at least five relations.
We have found the missing relation, and proved that it can not be represented as an index theorem.
2. (in collaboration with N. Goncharuk)
The foliation defined by a generic vector field $v$ possesses infinitely many homologically independent complex limit cycles.
Though this result is known for at least 39 years, the proof involved rather long estimates on some integrals.
We provide a very short geometric proof of this statement, that works in some open subsets of a more natural class of polynomial foliations of $\mathbb{CP}^2$ with prescribed $\textbf{projective}$ degree.