We study the symplectic geometry of rationally connected -folds. The first result shows that rational connectedness is a symplectic deformation invariant in dimension . If a rationally connected -fold is Fano or has Picard number , we prove that there is a nonzero Gromov–Witten invariant with two insertions being the class of a point. That is, is symplectic rationally connected. Finally we prove that many rationally connected -folds are birational to a symplectic rationally connected variety.
Zhiyu Tian. "Symplectic geometry of rationally connected threefolds." Duke Math. J. 161 (5) 803 - 843, 1 April 2012. https://doi.org/10.1215/00127094-1548398