Established Stabilizer-Induction Step

Definitions and mathematical statement

For the source's unspecified base field $k$ and matrix size $n$, write $G=\mathrm{SL}_{n,k}$, the algebraic group of determinant-one matrices. Write $H^{\mathrm{der}}$ for the derived subgroup of the group $H$ appearing in the source. Neither $H$ nor $X$, nor their relationship, is defined in the supplied record.

Let $\mathsf D(Z)$ be a placeholder for the source's unspecified assertion of "density" for $Z$. This placeholder does not specify a subset, an ambient space, or a topology. With this explicit limitation, the extracted conditional assertion has the following schematic form: \[ \begin{gathered} \text{Conjecture 9.1},\\ \text{for every homogeneous }G\text{-space }Y \text{ with geometric stabilizer equal to }H^{\mathrm{der}},\\ \text{and every smooth compactification }\overline Y\text{ of }Y, \quad \mathsf D(\overline Y) \end{gathered} \quad\Longrightarrow\quad \mathsf D(X). \] Here a smooth compactification means a smooth proper variety containing the given space as a dense open subvariety. The equality of the geometric stabilizer with $H^{\mathrm{der}}$ reproduces the source wording; its interpretation requires missing information about $H$, its field of definition, and the relevant identification.

Unresolved formulation and status. The statement of Conjecture 9.1, the hypotheses on $k,n,H,X$, and the meaning of density are absent. Consequently, the display is a conditional template, not a fully specified mathematical proposition. The supplied verification note says that the official report proves this inductive step under its external Conjecture 9.1 and that the extracted conditional assertion is not a separate open problem. This must be retained alongside the supplied open label and the unrecoverable statement status; none is independently verified here.

Short English statement

The record asks whether Conjecture 9.1 allows density for all smooth compactifications of homogeneous $\mathrm{SL}_n$-spaces with geometric stabilizer $H^{\mathrm{der}}$ to imply density for $X$. Essential definitions and hypotheses are missing. The supplied verification note describes the conditional step as already proved in the report, so the record does not provide a recoverable separate open problem.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003769$, source code catalogue entry, title "Established Stabilizer-Induction Step." No individual proposer is supplied.
• Supplied report citation: Mini-Workshop: Arithmetic Geometry and Symmetries around Galois and Fundamental Groups (2019), Oberwolfach Reports, pp. 1021--1080, DOI 10.4171/owr/2018/17. https://doi.org/10.4171/owr/2018/17. The record also calls this the "OWR 2018 report" and supplies proposed year 2018; the differing year labels are unresolved.
• Supplied literature source: Harpaz--Wittenberg, "homogeneous spaces." https://arxiv.org/abs/1802.09605. No more precise bibliographic attribution or relevant theorem identifier is supplied.
• Literature assessment dated 2026-08-21: supplied dated curation, not independently verified. It labels the status open and states that the proposed inductive step is explicitly conditional on Conjecture 9.1 and that later homogeneous-space results do not establish that conjecture in the recorded generality. The background describes this as literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: the official report proves the inductive step under its external Conjecture 9.1; the extracted conditional assertion is not a separate open problem. No separate clean statement is promoted, and the original record is retained publicly for provenance. This is supplied source-verification information, not a claim of independent source consultation.