Stability of the symplectomorphism group of rational surfaces

Abstract

We apply Zhang's almost Kähler Nakai-Moishezon theorem and Li-Zhang's comparison of $J$-symplectic cones to establish a stability result for the symplectomorphism group of a rational $4$-manifold $M$ with Euler number up to $12$. As a corollary, we also derive a stability result for the space of embedded symplectic balls in $M$. A noteworthy feature of our approach is that we systematically explore various spaces and groups associated to a symplectic cohomology class $u$ rather than with a single symplectic form $\omega$. To this end, we prove a weaker version of the tamed $J$-inflation procedures of D. McDuff and O. Buse that fixes a gap in their original formulations.

31 pages; v2, added stronger results on space of ball embeddings; v3, improved exposition. Final version

References in corpus (4)

• The topology of the space of symplectic balls in rational 4-manifolds
• The homotopy type of the space of symplectic balls in rational ruled 4-manifolds
• Symplectic $(-2)$-spheres and the symplectomorphism group of small rational 4-manifolds, II
• Topology of symplectomorphism groups and ball-swappings

Cited by in corpus (3)

• Symplectic $(-2)$-spheres and the symplectomorphism group of small rational 4-manifolds, II
• From Smooth to Almost Complex
• Symplectic isotopy on non-minimal ruled surfaces