Refined Donaldson–Thomas Meaning of Block–Göttsche Polynomials

Definitions and mathematical statement

Let $S$ denote a surface in the class intended by the source, let $\omega_S$ denote its canonical line bundle, and write \[ K_S:=\operatorname{Tot}(\omega_S). \] The source describes $K_S$ as a Calabi--Yau threefold. Write $\operatorname{BPS}_{\beta_S}^{S}(q)$ for the generalized Block--Göttsche polynomial attached in the source to $S$ and the class label $\beta_S$, with polynomial variable $q$.

Conjectural target. For each surface $S$ and class $\beta_S$ permitted by the source's construction, do the polynomials \[ \operatorname{BPS}_{\beta_S}^{S}(q) \] admit an interpretation in refined Donaldson--Thomas theory of $K_S$? The supplied literature assessment describes the intended interpretation as an identification with refined BPS invariants of $K_S$.

Unresolved scope and definitions. The supplied record does not specify the permitted surfaces, the admissible classes $\beta_S$, the construction or normalization of these polynomials, or the precise refined Donaldson--Thomas invariant and variable convention to be compared with them. Thus the preceding quantification refers to an unspecified source-defined range; it cannot be replaced here by an explicit universal quantifier over a determined class of surfaces and classes. A precise invariant equality requires those missing definitions.

Reported special case and qualification. The original excerpt says that degeneration arguments prove "this conjecture" when $S$ is a del Pezzo surface, described there as the case $n\leq 8$. It then states a theorem attributed to Argüz--Bousseau: \[ \operatorname{BPS}_{\beta_S}^{S}(-1)=W_{\beta_S} \qquad\text{for every }n\leq 8. \] Here $n$ and $W_{\beta_S}$ are retained as source notation: their definitions, admissible classes, and any real-geometric or enumerative choices are absent from the supplied excerpt. The relationship between this specialization identity and the full refined Donaldson--Thomas interpretation is not specified. In particular, the excerpt's claim to prove "this conjecture" in the del Pezzo case must be reconciled with the supplied dated assessment that the proposed identification remains conjectural and only related special evidence is established.

Short English statement

The question asks whether generalized Block--Göttsche polynomials associated with a surface $S$ can be identified with suitable refined Donaldson--Thomas BPS invariants of the total space of its canonical line bundle. The permitted surfaces, classes, and precise invariants are unspecified. The source also reports an Argüz--Bousseau identity at $q=-1$ for $n\leq 8$, but its connection to the full conjecture requires clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30005942$, source problem code catalogue entry, title: Refined Donaldson--Thomas Meaning of Block--Göttsche Polynomials. The supplied record assigns proposed year 2024 and names no proposer. Its status is recorded as open, and its statement status as corrected_verified; the accompanying claim of verification against the official report is supplied metadata, not an independent verification in this draft.
• Real and Logarithmic Enumerative Geometry (2024), Oberwolfach Reports, Report 23/2024, pp. 1285--1338, DOI 10.4171/owr/2024/23. Supplied report reference and literature source: https://doi.org/10.4171/owr/2024/23. The supplied original excerpt attributes the displayed specialization theorem to Argüz--Bousseau.
• BPS polynomials and Welschinger invariants. Supplied literature source: https://arxiv.org/abs/2506.02770.
• Supplied literature assessment, checked on 2026-08-22: the proposed identification with refined BPS invariants of the canonical-line-bundle Calabi--Yau remains conjectural, with only related special evidence established. This is supplied dated curation derived from the Oberwolfach material, not independently verified here; time-sensitive status and cited claims require independent checking.