Completed Masur–Veech Volumes of Odd Strata

Definitions and mathematical statement

A meromorphic quadratic differential on a compact connected Riemann surface $X$ of genus $g$ is a nonzero meromorphic section $q$ of $K_X^{\otimes 2}$, where $K_X$ is the canonical line bundle. If $\kappa=(k_1,\ldots,k_n)$ records all zero and pole orders of $q$, then \[ \sum_{i=1}^{n} k_i=4g-4. \] Positive orders describe zeros and negative orders describe poles. This recalls standard terminology only: the supplied record does not specify the permitted poles, genera, markings, or connected components. It also does not define "odd"; interpreting this as requiring every $k_i$ to be odd requires confirmation from the source.

The intended conjecture can be recorded only schematically. Let $\mathcal A$ stand for the source's unspecified collection of admissible odd quadratic-differential strata. For $\kappa\in\mathcal A$, write $\mathcal Q(\kappa)$ for the corresponding stratum, $V(\kappa)$ for its actual Masur--Veech volume, and $\widehat V(\kappa)$ for its completed Masur--Veech volume. Let $I(\kappa)$ index the adjacent-stratum contributions specified in the original source, with corresponding strata $\mathcal Q(\kappa_\alpha)$ and prescribed weights $w_{\kappa,\alpha}$. The requested identities have the form \[ \forall\,\kappa\in\mathcal A,\qquad \widehat V(\kappa) = V(\kappa)+\sum_{\alpha\in I(\kappa)} w_{\kappa,\alpha}\,V(\kappa_\alpha). \] The weights here represent fixed coefficients intended by the source, not coefficients whose existence is to be proved. The request includes the coefficients for the first strata mentioned in the original statement.

Unresolved formulation. The supplied record omits the displayed identities and listed coefficients to which its original statement refers. It supplies no definition of completed volume, no volume normalization, no admissibility criterion, and no specification of adjacency or of $I(\kappa)$, $\kappa_\alpha$, and $w_{\kappa,\alpha}$. Consequently, the display above is a template for the intended universal assertion, not a fully specified conjecture. These missing data must be recovered before a precise mathematical statement can be completed.

Short English statement

The requested conjecture says that every admissible odd quadratic-differential stratum has completed Masur--Veech volume equal to its actual volume plus prescribed weighted volumes of adjacent strata. The supplied record omits the defining formulas, admissibility conditions, normalization, and coefficients, so only a schematic statement is possible. Its partially solved status is supplied dated curation and has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004710$, title Completed Masur--Veech Volumes of Odd Strata, exact source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The supplied proposed year is 2021; no individual proposer is identified.
• Supplied source citation: Enumerative Geometry of Surfaces (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/28, pp. 1463--1518. https://doi.org/10.4171/owr/2021/28. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The report text and the referenced displayed coefficients are not included in the supplied record.
• Supplied literature source: Duryev--Goujard--Yakovlev, https://arxiv.org/abs/2502.13121. No title or theorem reference is supplied.
• Supplied literature source: Möller--Prado, https://arxiv.org/abs/2605.27611. No title or theorem reference is supplied.
• Supplied literature assessment dated 2026-08-22: the record labels the problem partially_solved and states that completed-volume formulas and the technically difficult two-singularity case have been proved, while the full conjectural equality and all stated adjacent-stratum weights are not established in general. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its repaired standalone formulation was checked against the official Oberwolfach report. These are supplied source-verification claims; this draft does not independently verify them or establish mathematical correctness.