Symplectic geometry of rationally connected threefolds

We study the symplectic geometry of rationally connected 3-folds. The first result shows that rational connectedness is a symplectic deformation invariant in dimension 3. If a rationally connected 3-fold X is Fano or has Picard number 2, we prove that there is a nonzero Gromov–Witten invariant with two insertions being the class of a point. That is, X is symplectic rationally connected. Finally we prove that many rationally connected 3-folds are birational to a symplectic rationally connected variety.