Chambers in the symplectic cone and stability of symplectomorphism group for ruled surface

TL;DR

This paper investigates the stability and topological structure of symplectomorphism groups on non-minimal ruled surfaces, revealing how embedded holomorphic curves influence stability and describing the colimit behavior as certain parameters grow.

Contribution

It extends previous work by analyzing symplectomorphism group stability via holomorphic curves and characterizes the topological colimit of these groups under specific blowup and area conditions.

Findings

Stability of symplectomorphism groups is governed by embedded J-holomorphic curves.

Identifies a topological colimit of symplectomorphism groups as the area ratio tends to infinity.

Constructs non-trivial symplectic mapping classes outside Dehn twists along Lagrangian spheres.

Abstract

We continue our previous work to prove that for any non-minimal ruled surface $(M,\omega)$, the stability under symplectic deformations of $\pi_0, \pi_1$ of $Symp(M,\omega)$ is guided by embedded $J$-holomorphic curves. Further, we prove that for any fixed sizes blowups, when the area ratio $\mu$ between the section and fiber goes to infinity, there is a topological colimit of $Symp(M,\omega_{\mu}).$ Moreover, when the blowup sizes are all equal to half the area of the fiber class, we give a topological model of the colimit which induces non-trivial symplectic mapping classes in $Symp(M,\omega) \cap \rm Diff_0(M),$ where $\rm Diff_0(M)$ is the identity component of the diffeomorphism group. These mapping classes are not Dehn twists along Lagrangian spheres.