Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

Hicks, Jeffrey

doi:10.2140/gt.2025.29.1909

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Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

Jeffrey Hicks

Geometry & Topology 29 (2025) 1909–1973

DOI: 10.2140/gt.2025.29.1909

Abstract

We say that a tropical subvariety $V \subset ℝ^{n}$ 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 $L_{V} \subset {(ℂ^{*})}^{n}$ . 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.

Keywords

tropical geometry, realizability, Lagrangian submanifolds, mirror symmetry

Mathematical Subject Classification

Primary: 14T20, 53D37

References

Publication

Received: 15 May 2022

Revised: 10 December 2023

Accepted: 12 January 2024

Published: 27 June 2025

Proposed: Leonid Polterovich

Seconded: David Fisher, Mark Gross

Authors

Jeffrey Hicks
School of Mathematics University of Edinburgh Edinburgh United Kingdom
https://www.maths.ed.ac.uk/~jhicks2

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Volume 29, issue 4 (2025)