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Rigidity and Realizability for Tropical Curves in Dimension 3

Jeff Hicks

TL;DR

The paper addresses unobstructedness for Lagrangian lifts of tropical curves in dimension 3 and its implications for realizability on the mirror side. It develops an unobstructedness criterion on the A-side from cyclic $A_\infty$ structures via a limiting loop grading $\mathrm{Grd}_\infty$, and then shows that rigid tropical curves with pair-of-pants decompositions yield unobstructed Lagrangian lifts $L_V$ in a tropical abelian threefold. Under a homological mirror symmetry framework, these unobstructed lifts imply a $B$-realization $Y_V$ in the mirror $X^B$, connecting tropical rigidity to algebraic realizability. The work also provides concrete examples illustrating rigidity, stabilization, and the calculation of relevant cohomology/functions that govern unobstructedness, thereby extending realizability criteria beyond previously known cases.

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 $X^B\to Q$.

Rigidity and Realizability for Tropical Curves in Dimension 3

TL;DR

The paper addresses unobstructedness for Lagrangian lifts of tropical curves in dimension 3 and its implications for realizability on the mirror side. It develops an unobstructedness criterion on the A-side from cyclic structures via a limiting loop grading , and then shows that rigid tropical curves with pair-of-pants decompositions yield unobstructed Lagrangian lifts in a tropical abelian threefold. Under a homological mirror symmetry framework, these unobstructed lifts imply a -realization in the mirror , connecting tropical rigidity to algebraic realizability. The work also provides concrete examples illustrating rigidity, stabilization, and the calculation of relevant cohomology/functions that govern unobstructedness, thereby extending realizability criteria beyond previously known cases.

Abstract

We present an unobstructedness criterion for Lagrangian threefolds using the -class associated with the boundary of a pseudoholomorphic disk. As an application, let be a Lagrangian torus fibration whose base is a tropical abelian threefold. Given a rigid tropical curve with a pair-of-pants decomposition, we prove that the Lagrangian lift is unobstructed. Provided that an appropriate homological mirror symmetry statement holds, this implies the existence of a realization in the mirror abelian threefold .

Paper Structure

This paper contains 18 sections, 7 theorems, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background: deformation problems in tropical and symplectic geometry
  3. Summary of Results
  4. Limitations of our Approach
  5. Acknowledgements
  6. A-side: Unobstructedness from Loop Grading
  7. Cyclic Structures in Floer Cohomology
  8. Properties of the Fukaya Algebra
  9. Summary of Construction of Product Structure
  10. Divisor axiom for the curvature term in dimension 3
  11. An unobstructedness criterion
  12. Unobstructedness from rigidity
  13. Constraints on the loop grading of disks
  14. First cohomology of Lagrangian lifts of tropical subvarieties
  15. Local model: the pair of pants
  16. ...and 3 more sections

Key Result

Theorem A

Let $\frac{1}{2}\dim_\mathbb R(X)=3$, and let $L\subset X$ be a graded oriented compact Lagrangian submanifold. Let $\{J_E\}$ be a sequence of almost complex structures so that the limit loop grading group $\mathop{\mathrm{Grd}}\nolimits_\infty(L, \{J_E\})\subset H_1(X)$ (see def:filteredLoopGrading

Figures (4)

  • Figure 1: Pseudoholomorphic disks with boundary on a tropical Lagrangian submanifold must tropicalize in the large complex structure limit.
  • Figure 2: A rigid tropical curve in a tropical abelian surface.
  • Figure 3: Stabilization preserves rigidity of tropical curves.
  • Figure 4: A tropical curve in a tropical abelian torus. The RGB cube is the unit cube drawn for reference. The dotted lines denote the boundary of the fundamental domain of the abelian torus.

Theorems & Definitions (26)

  • Theorem A
  • Definition 1.1
  • Theorem B
  • Conjecture 1.2
  • Conjecture 1.3
  • Remark 1.4
  • Theorem 2.1: fukaya2010cyclic
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 16 more