Symplectic invariance of rational surfaces on Kähler manifolds

Abstract

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.

37 pages. Revised draft: added statements about Koll{á}r's conjecture. Incorporated many suggestions of readers