Immersed two-spheres and SYZ with Application to Grassmannians

We develop a Floer theoretical gluing technique and apply it to deal with the most generic singular fiber in the SYZ program, namely the product of a torus with the immersed two-sphere with a single nodal self-intersection. As an application, we construct immersed Lagrangians in $\mathrm{G}\mathrm{r}(2,{\mathbb{C}}^n)$ and $\mathrm{O}\mathrm{G}(1,{\mathbb{C}}^5)$ and derive their SYZ mirrors. It recovers the Lie theoretical mirrors constructed by Rietsch. It also gives an effective way to compute stable disks (with non-trivial obstructions) bounded by immersed Lagrangians.