Orbifold Lifts of Deligne Mostow Forgetful Maps

Definitions and mathematical statement

Let $\mu$ and $\nu$ be Deligne--Mostow weight systems satisfying the source's integrality condition. Write $U_\mu$ and $U_\nu$ for the corresponding open moduli spaces of weighted configurations, and let $F:U_\mu\to U_\nu$ be the stated forgetful map. In the customary setting these configurations are marked points on ${\mathbb{P}}^1$ modulo projective transformations, with weight-dependent complex-hyperbolic metrics; the precise weight and labeling conventions are not supplied.

Let $\widehat U_\mu$ and $\widehat U_\nu$ be the relevant metric completions. Determine when $F$ extends to a map \[ \overline F:\widehat U_\mu\longrightarrow\widehat U_\nu, \qquad \overline F|_{U_\mu}=F. \] When it does, determine when the extension lifts to a map of the associated orbifolds. The source calls this equivalent to lifting to suitable torsion-free manifold covers. In that cover formulation the intended requirement is the existence of covers $q_\mu:\widetilde U_\mu\to\widehat U_\mu$, $q_\nu:\widetilde U_\nu\to\widehat U_\nu$, and a map $\widetilde F$ such that \[ q_\nu\circ\widetilde F=\overline F\circ q_\mu. \] The completion, regularity of the extended map, orbifold structures, and allowed covers must be specified before these criteria are precise. In particular, the asserted orbifold/cover equivalence is retained as source wording rather than proved for unspecified structures.

Short English statement

Classify when Deligne--Mostow forgetful maps extend across metric-completion strata and when those extensions respect orbifold structures or lift to suitable torsion-free covers. The weight, metric and covering conventions are missing from the extraction.

Sources

• Local UnsolvedMath registry, permanent ID $30001077$, source code catalogue entry, "Orbifold Lifts of Deligne Mostow Forgetful Maps." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Geometry and Arithmetic around Hypergeometric Functions (2009), Oberwolfach Reports, pp. 2557--2620, DOI https://doi.org/10.4171/owr/2008/45; supplied label "OWR Report 45."
• Supplied literature triage, literature_checked_at: 2026-08-21: labeled open; says the passage seeks a general extension criterion for the relevant hypergeometric orbifold maps and no resolving result for the general existence and lifting question was found. This is supplied dated curation, not independently verified.