Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds

Hicks, Jeffrey¹

¹School of Mathematics, University of Edinburgh, Edinburgh, United Kingdom

Geometry & topology, Tome 29 (2025) no. 4, pp. 1909-1973

Résumé

We say that a tropical subvariety V ⊂ ℝn 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 LV ⊂ (ℂ∗)n. We prove that whenever LV has well-defined Floer cohomology, we can find for each point of V a Lagrangian torus brane whose Lagrangian intersection Floer cohomology with LV is nonvanishing. Assuming an appropriate homological mirror symmetry result holds for toric varieties, it follows that whenever LV 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.

Détail

DOI : 10.2140/gt.2025.29.1909

Keywords: tropical geometry, realizability, Lagrangian submanifolds, mirror symmetry

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