Quantization near Singular Faces of Delzant Polytopes

Definitions and mathematical statement

For a positive integer $n$, a Delzant polytope $P\subset ({\mathbb{R}}^n)^*$ is a full-dimensional compact convex polytope that is simple, rational, and smooth: exactly $n$ facets meet at each vertex, and their primitive inward normal vectors form a basis of the lattice ${\mathbb{Z}}^n$. An edge is a one-dimensional face; a vertex is a zero-dimensional face. These definitions explain the terminology and do not specify a dimension for the supplied problem.

In a Hamiltonian torus interpretation, one would choose a symplectic manifold $(M,\omega)$ with an effective Hamiltonian action of $T^n=({\mathbb{R}}/{\mathbb{Z}})^n$ and a moment map \[ \mu:M\longrightarrow({\mathbb{R}}^n)^*,\qquad \mu(M)=P. \] For a face $F\subseteq P$, the corresponding locus is $\mu^{-1}(F)$. This is a possible framework for interpreting the question, not additional data supplied by the record. An elliptic singular point refers to a singularity of the relevant integrable system or polarization on $M$, rather than an intrinsically defined singular point of the convex polytope. The record does not specify that system or polarization.

Source question. What is the behavior of geometric quantization at the edges and vertices, including singular elliptic points, of a Delzant polytope?

In the moment-map interpretation above, this asks for a description of geometric quantization associated with the loci $\mu^{-1}(E)$ and $\mu^{-1}(v)$, where $E$ is an edge and $v$ is a vertex, including the treatment of elliptic singularities. The source does not determine whether the desired description concerns local quantization, contributions to a global quantum space, behavior of polarized sections, or a limiting construction near these loci. It also does not specify the prequantization data, polarization, treatment of singular leaves, or use of a half-form correction. Consequently, the supplied question cannot yet be expressed as a uniquely determined quantified mathematical assertion. In particular, no vanishing, dimension, or extension claim is asserted here.

The supplied background also refers to a general singular setting, without defining its relationship to the Delzant setting. This qualification remains unresolved.

Short English statement

The question asks how geometric quantization treats the edges and vertices of a Delzant moment polytope, including elliptic singularities in the associated geometric system. The quantization construction and the intended meaning of its behavior are unspecified, so this remains an explicitly qualified research question requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001710$, title Quantization near Singular Faces of Delzant Polytopes, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. The supplied proposed year is 2011; no individual proposer is identified.
• Geometric Quantization in the Non-compact Setting (2011), Oberwolfach Reports, pp. 425--521, DOI 10.4171/owr/2011/09. Supplied source URL and literature-source link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/09. This report citation is supplied provenance; it has not been independently inspected for this draft.
• Supplied literature assessment dated exactly 2026-08-21: the record labels the problem partially_solved and states that Hamilton and the report treat toric and low-dimensional nondegenerate singularities, while the question about edges and vertices in the general singular setting is not fully resolved. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The source explicitly requests independent verification of time-sensitive status and cited claims. No separate Hamilton reference is supplied.
• The registry marks the statement as exact and reports a manual review finding the extraction self-contained, mathematically coherent, and free of LaTeX or report-layout artifacts. Those are supplied verification claims, not independent verification of the formulation or its mathematical status in this draft.