Semicomplete meromorphic vector fields on complex surfaces (Q2898922)

The authors study meromorphic vector fields on complex surfaces such that the vector field obtained when they are restricted to the complement of their locus of poles is semicomplete. Semicomplete means that, viewed as ordinary differential equations, their solutions are single-valued in their maximal definition domain. The main result in the paper can be thought as a local minimal model statement for semicomplete meromorphic vector fields. In the paper it is proved that for such vector fields, up to a birational transformation, a compact connected component of the curve of poles is either a rational or an elliptic curve whose self-intersection vanishes or it has the combinatorics of a singular fiber of an elliptic fibration. This result is also globalized by proving that, 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: it is globally holomorphic, it has a non-trivial meromorphic first integral or it preserves a fibration.