Poincar\'e-Bendixson theorems for meromorphic connections and homogeneous vector fields

We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincar'e-Bendixson theorem describing recurrence properties and $\omega$-limit sets of geodesics for a meromorphic connection on ${\mathrm{\P }}^1(\text{C})$. We then show how to associate to a homogeneous vector field $Q$ in ${\mathbb{C}}^n$ a rank 1 singular holomorphic foliation $\mathcal{F}$ of ${\mathrm{\P }}^{n-1}(\text{C})$ and a (partial) meromorphic connection ${\mathrm{\nabla }}^o$ along $\text{ca}F$ so that integral curves of $Q$ are described by the geodesic flow of ${\mathrm{\nabla }}^o$ along the leaves of $\text{ca}F$, which are Riemann surfaces. The combination of these results yields powerful tools for a detailed study of the dynamics of homogeneous vector fields. For instance, in dimension two we obtain a description of recurrence properties of integral curves of $Q$, and of the behavior of the geodesic flow in a neighbourhood of a singularity, classifying the possible singularities both from a formal point of view and (for generic singularities) from a holomorphic point of view. We also get examples of unexpected new phenomena, we put in a coherent context scattered results previously known, and we obtain (as far as we know for the first time) a complete description of the dynamics in a full neighbourhood of the origin for a substantial class of 2-dimensional holomorphic maps tangent to the identity. Finally, as an example of application of our methods we study in detail the dynamics of quadratic homogeneous vector fields in ${\text{C}}^2$.