Moment-Map Images as Duistermaat–Heckman Supports

Definitions and mathematical statement

The supplied record does not specify the geometric data on which the connection and moment map are defined. To display the mathematical target without supplying missing hypotheses, let $M$ denote the unspecified domain of the moment map, let $V$ denote its unspecified topological target, and let $\mathcal C$ denote the unspecified class of admissible connections. For $\nabla\in\mathcal C$, write $\Phi_\nabla\colon M\to V$ for the corresponding moment map. Let $\mu_{\mathrm{DH}}$ denote the signed Duistermaat--Heckman measure referred to in the supplied literature assessment.

For a signed measure $\mu$ on a topological space $V$, use the support convention \[ \operatorname{supp}(\mu)=\left\{v\in V: \mu|_U\ne 0\text{ for every open neighborhood }U\text{ of }v\right\}. \] Here $\mu|_U\ne0$ means that the restriction is not the zero measure, rather than merely that $\mu(U)\ne0$.

The recorded question is schematically: \[ \text{Does there exist }\nabla\in\mathcal C\text{ such that } \Phi_\nabla(M)=\operatorname{supp}(\mu_{\mathrm{DH}})? \] The equality concerns the image itself, without replacing it by its closure. The record does not specify whether $\mu_{\mathrm{DH}}$ varies with $\nabla$; if it does, the right-hand side must use the measure associated with the same connection. The underlying manifold, symmetry action, bundle, admissibility conditions on connections, construction of the moment map and measure, and quantification over the underlying geometric data are absent. Thus this is an existential target with unresolved hypotheses, not a complete universally quantified mathematical statement.

Short English statement

Can an admissible connection be chosen so that its moment-map image equals the support of the signed Duistermaat--Heckman measure? The supplied record leaves the geometric setting and admissible connections unspecified, so the question remains schematic.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003073$, source problem code catalogue entry, title: Moment-Map Images as Duistermaat--Heckman Supports. The record assigns the year 2016 and names no proposer.
• Supplied source citation: Mini-Workshop: PBW Structures in Representation Theory (2016), Oberwolfach Reports, pp. 615--651, DOI 10.4171/owr/2016/13. Supplied source and literature link: https://doi.org/10.4171/owr/2016/13. The background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Supplied literature reference attributed to Paradan--Vergne: https://archive.intlpress.com/site/pub/files/_fulltext/journals/acta/2017/0218/0001/ACTA-2017-0218-0001-a003.pdf. The record does not supply a full bibliographic citation for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and attributes to Paradan and Vergne the unresolved connection-choice question and examples in which equality fails for a chosen connection. This is supplied dated curation derived from the OWR material, not independently verified here; the record expressly calls for independent verification of time-sensitive status and cited claims.