Immersed two-spheres and SYZ with Application to Grassmannians

2018/05/29 by Hansol Hong, Hong, Hansol, Yoosik Kim +3 · 2 citations
Mathematics · Physics and Astronomy · #53D37 #53D40 #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1805.11738

openalex publication_date 2018/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Gr(2,ℂⁿ) and OG(1,ℂ⁵) 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.

Citations

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Floer homology of Lagrangians in clean intersection
• Floer cohomology of immersed Lagrangian spheres in smoothings of A_N surfaces

Cited by

• T-equivariant disc potentials for toric Calabi-Yau manifolds
• Cyclic group actions on Fukaya categories and mirror symmetry

Related

• Refined disk potentials for immersed Lagrangian surfaces
• Homological Mirror Symmetry for local SYZ singularities
• T-equivariant disc potential and SYZ mirror construction
• Gluing Localized Mirror Functors
• Exact, graded, immersed Lagrangians and Floer theory