Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

Abstract

We say that a tropical subvariety $V\subset \mathbb R^n$ is $B$-realizable if it can be lifted to an analytic subset of $(\Lambda^*)^n$. When $V$ is a smooth curve or hypersurface, there always exists a Lagrangian submanifold lift $L_V\subset (\mathbb C^*)^n$. 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 non-vanishing. 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 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.