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

Abstract

We continue our previous work to prove that for any non-minimal ruled surface (M,ω), the stability under symplectic deformations of π0,π1 of Symp(M,ω) is guided by embedded J-holomorphic curves. Further, we prove that for any fixed sizes blowups, when the area ratio μ between the section and fiber goes to infinity, there is a topological colimit of Symp(M,ωμ). 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,ω)∩Diff0(M), where Diff0(M) is the identity component of the diffeomorphism group. These mapping classes are not Dehn twists along Lagrangian spheres.