Semicomplete vector fields on non-Kähler surfaces

Abstract

We investigate semicomplete meromorphic vector fields on complex surfaces, those where the solutions of the associated ordinary differential equations have no multivaluedness. We prove that if a non-Kähler compact complex surface has such a vector field, then, up to a bimeromorphic transformation, either the vector field is holomorphic, has a first integral or preserves a fibration. This extends previous results of Rebelo and the author to the non-Kähler setting.