Displaceability of Lagrangian Toric Fibers

Definitions and mathematical statement

Let $(M,\omega,T)$ be a symplectic toric manifold as in the supplied record: $T$ is a torus acting effectively and Hamiltonianly on the symplectic manifold $(M,\omega)$, with $\dim M=2\dim T$. Write $\mathfrak t$ for the Lie algebra of $T$, let $\Phi:M\to\mathfrak t^*$ be a moment map, and let $\Delta=\Phi(M)$ be the moment polytope specified in the record. For every $u\in\operatorname{int}\Delta$, put \[ L_u=\Phi^{-1}(u), \] the Lagrangian toric fiber over $u$. Here the interior is taken in $\mathfrak t^*$.

A fiber $L_u$ is Hamiltonian displaceable if there exists a Hamiltonian symplectomorphism $\varphi\in\operatorname{Ham}(M,\omega)$ such that \[ \varphi(L_u)\cap L_u=\varnothing. \] A Hamiltonian symplectomorphism is a time-one map of a smooth time-dependent Hamiltonian flow defined throughout the time interval $[0,1]$. If noncompact manifolds are included, the required support convention for the Hamiltonians must additionally be specified.

Classification problem. For each symplectic toric manifold in the intended class, determine the set \[ \mathcal D(M,\omega,\Phi) =\left\{u\in\operatorname{int}\Delta: \exists\varphi\in\operatorname{Ham}(M,\omega),\quad \varphi(L_u)\cap L_u=\varnothing\right\}. \] Equivalently, characterize precisely which interior toric fibers are Hamiltonian displaceable. The complementary fibers are Hamiltonian nondisplaceable, meaning that their intersection with every Hamiltonian image is nonempty. Boundary fibers are outside the stated problem.

Unresolved scope qualification. The record does not explicitly specify compactness, connectedness, or the absence of boundary for $M$. If the conventional compact, connected, boundaryless meaning of symplectic toric manifold is intended, the statement above is to be read in that class. The record does not establish an extension to a broader class; for a noncompact interpretation, the manifold class and Hamiltonian support convention require clarification.

Short English statement

For each symplectic toric manifold in the intended class, classify the interior moment-map fibers that a Hamiltonian symplectomorphism can move completely away from themselves. The source leaves the precise manifold and, if relevant, Hamiltonian support conventions implicit. Its partially solved designation is supplied dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001129$, title Displaceability of Lagrangian Toric Fibers, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2009 and names no individual proposer.
• Supplied report citation: Toric Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2009/01, pp. 5--74. This is also the literature-source entry labeled "original OWR report": https://doi.org/10.4171/owr/2009/01. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source entry labeled "Lagrangian Floer theory on toric manifolds": https://arxiv.org/abs/0802.1703.
• Supplied literature-source entry labeled "McDuff probes": https://arxiv.org/abs/0904.1686.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that Floer-theoretic potentials and probe constructions classify many displaceable and nondisplaceable toric fibers, while no complete classification for arbitrary symplectic toric manifolds is known. This is supplied dated curation derived from the OWR material, not independently verified here; its time-sensitive status and cited claims require independent checking.
• The supplied registry labels the statement corrected_verified, explaining that the displaceability problem is complete and that Hamiltonian-image, intersection, and empty-set notation was normalized. This records the supplied editorial assessment only.