Poincaré–Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields

We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincaré–Bendixson theorem describing recurrence properties and ω-limit sets of geodesics for a meromorphic connection on P1(C). We then show how to associate to a homogeneous vector field Q in Cn a rank 1 singular holomorphic foliation F of Pn−1(C) and a (partial) meromorphic connection ∇o along F so that integral curves of Q are described by the geodesic flow of ∇o along the leaves of 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 neighborhood 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 neighborhood of the origin for a substantial class of 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 C2.