Real Orbit Counts on Wonderful Varieties

Definitions and mathematical statement

The supplied question concerns the real points of a wonderful variety. The record does not specify all hypotheses on the group or explain the word "canonical." The following gives a conditional interpretation using the usual complex algebraic terminology; these conventions require comparison with the cited report.

Under this interpretation, let $G$ be a connected complex reductive algebraic group. A wonderful $G$-variety is a smooth, complete, irreducible complex algebraic $G$-variety $X$ with an open dense $G$-orbit whose complement is a union of finitely many smooth $G$-stable prime divisors $D_1,\ldots,D_r$. These divisors have simple normal crossings and nonempty total intersection, and the $G$-orbit closures are precisely their partial intersections, with the empty intersection interpreted as $X$. These are conventional definitions, not additional hypotheses verified from the report.

Let $\sigma:G\to G$ be an antiholomorphic involutive group automorphism, interpreted as a real form of $G$. Let $\mu:X\to X$ be a real structure, meaning an antiholomorphic involution, satisfying \[ \mu(g\cdot x)=\sigma(g)\cdot\mu(x) \qquad(g\in G,\ x\in X). \] The source further requires $\mu$ to be the canonical such structure. Its construction, existence hypotheses, and any intended uniqueness assertion are not supplied and remain unresolved qualifications.

Define \[ X({\mathbb{R}})=\{x\in X:\mu(x)=x\},\qquad G^\sigma=\{g\in G:\sigma(g)=g\},\qquad G_0^\sigma=(G^\sigma)^\circ, \] where the identity component is taken in the usual real Lie group topology. Equivariance makes $X({\mathbb{R}})$ invariant under $G_0^\sigma$. For $x,y\in X({\mathbb{R}})$, write \[ x\sim y\quad\Longleftrightarrow\quad \text{there exists }g\in G_0^\sigma\text{ with }y=g\cdot x. \] Orbit-count question. For each wonderful $G$-variety with the canonical $\sigma$-equivariant real structure intended by the source, determine exactly \[ N(G,X,\sigma,\mu) =\#\left(G_0^\sigma\backslash X({\mathbb{R}})\right) =\#\left(X({\mathbb{R}})/{\sim}\right). \] The record specifies no particular variety or numerical input, so this is a general determination problem. No nonemptiness assumption on $X({\mathbb{R}})$ is supplied; the displayed cardinality is zero when $X({\mathbb{R}})$ is empty. The intended restrictions on $G$, $\sigma$, and the canonical structure must be recovered before this conditional formulation can be treated as a precise transcription of the report.

Short English statement

Determine the exact number of orbits of the identity component of the real fixed-point group on the real locus of a wonderful variety carrying the source's canonical equivariant real structure. The precise group hypotheses and meaning of "canonical" are missing from the supplied record and require review. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002024$, exact source problem code catalogue entry, title "Real Orbit Counts on Wonderful Varieties," in the collection "Oberwolfach Reports Open Problems." No individual proposer is named; the supplied proposed year is 2012.
• Supplied primary-source citation: Mini-Workshop: Generalizations of Symmetric Spaces (2013), Oberwolfach Reports, pp. 1107--1148, DOI 10.4171/owr/2012/18. Supplied source URL: https://doi.org/10.4171/owr/2012/18. The citation's parenthetical year 2013 differs from the supplied proposed year 2012 and the year appearing in the DOI; this discrepancy is unresolved.
• Supplied literature source, "Oberwolfach report": https://doi.org/10.4171/OWR/2012/18.
• Supplied literature source, "Kac--Moody symmetric spaces": https://arxiv.org/abs/1702.08426.
• Supplied literature source, "Köhl--Vock": https://doi.org/10.1515/advgeom-2025-0036.
• Supplied literature-check date: 2026-08-21. This is supplied dated curation, not independently verified. Its assessment labels the problem open and states that subsequent structural work does not provide a general exact orbit count. The background describes this assessment as literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that the official report confirms the orbit-count question and supplies definitions omitted by extraction. This is a supplied verification claim; the report's defining passages are not reproduced in the record and have not been independently checked here.