Geometric Quotients of K-Sheets

Definitions and mathematical statement

The supplied record does not specify the ground field, the algebraic group $K$, its representation space, or the hypotheses under which the question is intended. The following definitions give a conditional formulation; their applicability to the source setting requires review.

Suppose that $K$ is an algebraic group acting on a variety $V$ over a field $k$. For an integer $d\geq 0$, set \[ V_{(d)}=\{x\in V:\dim(K\cdot x)=d\}. \] Under the usual orbit-dimension interpretation, a $K$-sheet is an irreducible component $S$ of some $V_{(d)}$, equipped with its reduced locally closed variety structure. This interpretation, including the intended ambient variety $V$, is not specified in the supplied record and must be confirmed.

A geometric quotient of a $K$-stable sheet $S$ means a surjective $K$-invariant morphism of varieties \[ q:S\longrightarrow Q \] whose geometric fibers are single $K$-orbits, for which $Q$ has the quotient topology, and for which pullback identifies regular functions on each open subset $U\subseteq Q$ with invariant regular functions on its inverse image: \[ \mathcal O_Q(U)\cong\mathcal O_S(q^{-1}(U))^K. \] The intended category of quotients and any additional requirements over the ground field remain unspecified.

Question. In the setting intended by the cited report, does every $K$-sheet admit a geometric quotient by $K$? If such a quotient exists, can it be described using the $p$-Slodowy slice?

The supplied record neither defines the $p$-Slodowy slice nor specifies what constitutes a description of the quotient using it. Consequently, the second question cannot be sharpened into a particular isomorphism, slice construction, or parametrization from the supplied evidence alone. No such construction is asserted here.

Short English statement

The question asks whether every $K$-sheet has a geometric quotient by $K$, and whether an existing quotient can be described using the $p$-Slodowy slice. The group, ambient space, field, and slice construction need source-level clarification. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001460$, title Geometric Quotients of K-Sheets, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Algebraic Groups (2010), Oberwolfach Reports, pp. 1101--1163, DOI 10.4171/owr/2010/19. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2010/19. The record attributes the question to this workshop report; no individual proposer is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the geometric-quotient question is explicitly posed in the report, while cited results establish equidimensionality and smoothness in cases, rather than a general quotient or a $p$-Slodowy description. This is supplied dated curation, not independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. These are supplied verification claims; no source visit or independent verification is asserted in this draft.