Rigidity and Realizability for Tropical Curves in Dimension 3

TL;DR

This paper establishes criteria for the unobstructedness of Lagrangian threefolds in symplectic geometry, linking tropical geometry and mirror symmetry to demonstrate the existence of certain geometric realizations.

Contribution

It introduces an unobstructedness criterion for Lagrangian threefolds and connects tropical curves with mirror symmetry in dimension three.

Findings

Unobstructedness criterion for Lagrangian threefolds

Existence of Lagrangian lifts for rigid tropical curves

Implication for mirror symmetry and realizations in mirror abelian threefolds

Abstract

We present an unobstructedness criterion for Lagrangian threefolds $L\subset X^A$ using the $H_1(L)$-class associated with the boundary of a pseudoholomorphic disk. As an application, let $X^A\to Q$ be a Lagrangian torus fibration whose base $Q$ is a tropical abelian threefold. Given $V\subset Q$ a rigid tropical curve with a pair-of-pants decomposition, we prove that the Lagrangian lift $L_V\subset X^A$ is unobstructed. Provided that an appropriate homological mirror symmetry statement holds, this implies the existence of a realization $Y_V$ in the mirror abelian threefold $X^B\to Q$.