Compactifying Lagrangian Lifts of Tropical Hypersurfaces

Definitions and mathematical statement

Let $P$ denote the toric moment polytope in the supplied statement, let $P^{\circ}$ be its interior, and let $\Lambda\subseteq P$ be the tropical hypersurface under consideration. The stated boundary condition is \[ \Lambda\cap\partial P\ne\varnothing. \] The source does not specify the dimension, regularity of $\Lambda$, its permitted intersections with faces of $P$, or the construction of its Lagrangian lift.

To describe the intended compactification question in a smooth symplectic ambient space, write $(X^{2n},\omega)$ for such a toric space with moment map $\mu:X\to P$, and write $j:L^{\circ}\hookrightarrow\mu^{-1}(P^{\circ})$ for a specified smooth embedded Lagrangian lift, when the intended construction supplies one. Here the Lagrangian condition means \[ \dim L^{\circ}=n,\qquad j^{*}\omega=0. \] The notation $L^{\circ}$ refers to the portion of the proposed lift over the interior of the polytope; the source does not define this lift or its precise relation to $\Lambda$.

Under the interpretation that compactification means an embedded extension of this specified lift, the target is the existence of a compact smooth $n$-manifold $\overline L$, a smooth open embedding $\iota:L^{\circ}\hookrightarrow\overline L$ with dense image, and a smooth embedding $\overline j:\overline L\hookrightarrow X$, such that \[ \partial\overline L=\varnothing,\qquad \overline j\circ\iota=j,\qquad \overline j^{*}\omega=0. \] Thus the question asks whether the prescribed lift extends smoothly across the toric boundary to a compact Lagrangian submanifold without boundary.

This is a conditional formulation of the target, not a fully specified universal conjecture: the admissible toric spaces, tropical hypersurfaces, and lifts are absent from the supplied record. It is also unspecified whether one must extend a fixed lift or may choose or modify the lift near the boundary, and whether embedded rather than immersed compactification is intended. The background phrase "inside the toric polytope" does not identify the symplectic ambient space; the formulation above interprets the compactification as lying in $X$, whose moment image lies in $P$. These qualifications require clarification before a definitive quantified statement can be given.

Short English statement

For a tropical hypersurface meeting the boundary of a toric moment polytope, ask whether its Lagrangian lift extends to a compact smooth Lagrangian submanifold without boundary in the associated toric space. The source leaves the admissible hypersurfaces, boundary behavior, ambient space, and lift construction unspecified. The displayed formulation describes an embedded extension of a fixed lift; whether that interpretation is intended remains unresolved. The supplied open status is a dated curation label, not an independently verified conclusion.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30004141$, source problem code catalogue entry, title "Compactifying Lagrangian Lifts of Tropical Hypersurfaces." The supplied collection is Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2019.
• Supplied report citation: Tropical Geometry: new directions (2020), Oberwolfach Reports, pp. 1245--1308, DOI 10.4171/owr/2019/21. https://doi.org/10.4171/owr/2019/21. The supplied background attributes the problem to mathematical workshop discussions documented in Oberwolfach Reports.
• Supplied literature source: "Tropical geometry problem report." https://ems.press/content/serial-article-files/46800.
• Supplied literature source: "Tropical Lagrangians in toric del-Pezzo surfaces." https://doi.org/10.1007/s00029-020-00614-1.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports smooth families from tropical Lagrangian constructions in low-dimensional torus settings, while stating that a general smooth boundaryless compactification inside the toric polytope is not established. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The cited claims and time-sensitive status require independent verification.