2022/04/13 by J. Hicks, Hicks, Jeff · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematics and Applications #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2204.06432
openalex publication_date 2022/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We say that a tropical subvariety V⊂ \mathbb Rn is B-realizable if it can be lifted to an analytic subset of (Λ^*)n. When V is a smooth curve or hypersurface, there always exists a Lagrangian submanifold lift LV⊂ (\mathbb C^*)n. We prove that whenever LV has well-defined Floer cohomology, we can find for each point of V a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with LV is non-vanishing. Assuming an appropriate homological mirror symmetry result holds for toric varieties, it follows that whenever LV is a Lagrangian submanifold that can be made unobstructed by a bounding cochain, the tropical subvariety V is B-realizable. As an application, we show that the Lagrangian lift of a genus zero tropical curve is unobstructed, thereby giving a purely symplectic argument for Nishinou and Siebert's proof that genus-zero tropical curves are B-realizable. We also prove that tropical curves inside tropical abelian surfaces are B-realizable.