Immersed two-spheres and SYZ with Application to Grassmannians

Abstract

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 $\mathrm{Gr}(2,{\mathbb{C}}^n)$ and $\mathrm{OG}(1,{\mathbb{C}}^5)$ 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.

61 pages, 24 figures, comments are welcome; [v2] mirror construction for 2-plane Grassmannians in general dimension is added

References in corpus (4)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Toric degeneration and non-displaceable Lagrangian tori in $S^2\times S^2$
• Gluing Localized Mirror Functors
• Moduli of Lagrangian immersions with formal deformations

Cited by in corpus (9)

• Fukaya category of Grassmannians: rectangles
• On the Complex Affine Structures of SYZ Fibration of Del Pezzo Surfaces
• Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces
• $T$-equivariant disc potentials for toric Calabi-Yau manifolds
• Kähler Geometry of Quiver Varieties and Machine Learning
• Pseudotoric structures and special Lagrangian torus fibrations on certain flag varieties
• Family Floer mirror space for local SYZ singularities
• Maurer-Cartan deformation of Lagrangians
• Cyclic group actions on Fukaya categories and mirror symmetry