Towards the symplectic Graber-Harris-Starr theorems

A theorem of Graber, Harris, and Starr states that a rationally connected fibration over a curve has a section. We study an analogous question in symplectic geometry. Namely, given a rationally connected fibration over a curve, can one find a section which gives a non-zero Gromov-Witten invariant? We observe that for any fibration, the existence of a section which gives a non-zero Gromov-Witten invariant only depends on the generic fiber, i.e. a variety defined over the function field of a curve. Some examples of rationally connected fibrations with this property are given, including all rational surface fibrations. We also prove some results, which says that in certain cases we can "lift" Gromov-Witten invariants of the base to the total space of a rationally connected fibration.