Symplectic Singularities from Symplectic Reduction

Definitions and mathematical statement

The supplied question is: When does symplectic reduction produce a symplectic singularity?

Under the usual complex algebraic convention, a symplectic singularity is a normal complex algebraic variety $X$ whose smooth locus $X_{\mathrm{reg}}$ admits a regular two-form $\omega$ satisfying $\mathrm d\omega=0$, such that $\omega$ is nondegenerate at every point of $X_{\mathrm{reg}}$, and such that for every resolution of singularities $\pi:\widetilde X\to X$, the form $\pi^*\omega$, initially defined on $\pi^{-1}(X_{\mathrm{reg}})$, extends to a regular two-form on all of $\widetilde X$. The extension is not required to be nondegenerate.

With this convention as a possible interpretation of the target, the problem asks for conditions on symplectic reduction data under which the resulting space has this property. The source does not specify the reduction data or the category: it gives no ambient symplectic space, acting group, moment map, reduction value, or quotient construction. It also does not specify whether the form on the smooth locus must be the one induced by reduction, or whether normalization or another modification of the quotient is allowed. Consequently, the supplied question is a characterization problem with unresolved scope, rather than a fully specified universally quantified assertion. The definition above explains a conventional meaning of the target; the source does not establish that this particular convention is intended.

Short English statement

Find conditions ensuring that a symplectic reduction has symplectic singularities. In the conventional complex algebraic interpretation, this requires normality, a symplectic form on the smooth locus, and extension of that form across every resolution. The source leaves the reduction setting and the precise intended scope unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001346$, title Symplectic Singularities from Symplectic Reduction, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2009; no individual proposer is identified.
• Supplied report citation: Singularities (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/43, pp. 2405--2470. https://doi.org/10.4171/owr/2009/43. This is the supplied original-report reference; its contents have not been independently checked here.
• Supplied literature reference, labeled "symplectic reductions for polar representations": https://arxiv.org/abs/1604.04121.
• Supplied literature reference, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2009/43.
• The literature assessment dated exactly 2026-08-21 is supplied dated curation, not independently verified. It labels the problem partially solved and states that general symplectic reductions need not be symplectic singularities, that criteria are proved for polar representations and other classes, and that no general characterization is known. The supplied background describes this assessment as literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.