  Kollar and Ruan proved symplectic deformation invariance for uniruledness of
Kaehler manifolds. Zhiyu Tian proved the same for rational connectedness in
dimension < 4. Kollar conjectured this in all dimensions. We prove Kollar's
conjecture, as well as existence of a covering family of rational surfaces, for
all Kaehler manifolds that are symplectically deformation equivalent to G/P or
to a low degree complete intersection in such.
