Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

TL;DR

This paper explores the connection between tropical geometry and symplectic topology, showing that certain tropical subvarieties can be lifted to unobstructed Lagrangian submanifolds, with implications for mirror symmetry.

Contribution

It establishes conditions under which tropical subvarieties are realizable as unobstructed Lagrangian submanifolds, linking tropical geometry to Floer cohomology and mirror symmetry.

Findings

Lagrangian lifts of smooth curves and hypersurfaces can be unobstructed.

Genus zero tropical curves are proven to be B-realizable.

Tropical curves in abelian surfaces are B-realizable.

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.