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 $(M,\omega )$, 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 $Symp(M,\omega )\cap \mathrm{Dif}{\mathrm{f}}_0(\mathrm{M}),$ where $\mathrm{Dif}{\mathrm{f}}_0(\mathrm{M})$ is the identity component of the diffeomorphism group. These mapping classes are not Dehn twists along Lagrangian spheres.

25 pages. Comment welcome! arXiv admin note: text overlap with arXiv:2010.04616