Semicomplete meromorphic vector fields on complex surfaces

Abstract

We study semicomplete meromorphic vector fields on complex surfaces, that is, vector fields whose solutions are single-valued in restriction to the open set where the vector field is holomorphic. We show that, up to a birational transformation, a compact connected component of the curve of poles is either a rational or an elliptic curve of null self-intersection or it has the combinatorics of a singular fiber of an elliptic fibration. This result is then globalized by proving that, always up to a birational transformation, a semicomplete meromorphic vector field on a compact complex Kähler surface must satisfy at least one of the following conditions: to be globally holomorphic, to possess a non-trivial meromorphic first integral or to preserve a fibration. In particular, this extends the results established by Brunella for complete polynomial vector fields in the complex plane to the context of semicomplete ones.