Symplectic geometry of rationally connected threefolds

Abstract

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.