Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

We say that a tropical subvariety V ⊂ 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 L_V ⊂ (ℂ^∗)ⁿ. We prove that whenever L_V has well-defined Floer cohomology, we can find for each point of V a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with L_V is nonvanishing. Assuming an appropriate homological mirror symmetry result holds for toric varieties, it follows that whenever L_V 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-0 tropical curve is unobstructed, thereby giving a purely symplectic argument for Nishinou and Siebert’s proof that genus-0 tropical curves are B-realizable. We also prove that tropical curves inside tropical abelian surfaces are B-realizable.