Symplectic invariance of rational surfaces on K\"{a}hler manifolds

TL;DR

This paper proves Kollar's conjecture that symplectic deformation invariance of rational surfaces holds for all Kähler manifolds, extending previous results on uniruledness and rational connectedness.

Contribution

It establishes the existence of a covering family of rational surfaces on all Kähler manifolds symplectically equivalent to G/P or low degree complete intersections.

Findings

Proves Kollar's conjecture in all dimensions.

Shows existence of rational surface families on relevant Kähler manifolds.

Extends symplectic invariance results to broader classes of 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.