Stable-Pairs and Gromov–Witten Series for K3 Times Elliptic Curves

Definitions and mathematical statement

Let $S$ be a K3 surface and let $E$ be an elliptic curve. The supplied title identifies the intended geometry as the product $X=S\times E$. Write $F_{\mathrm{PT}}$ and $F_{\mathrm{GW}}$ for the stable-pairs and Gromov--Witten generating series, respectively, referred to in the supplied record, and write $F$ for its intended explicit correction term.

The supplied question, called an Igusa cusp-form identity in the record, is whether, for every pair $(S,E)$ satisfying the intended hypotheses, one has \[ F_{\mathrm{PT}}=F_{\mathrm{GW}}+F. \] Here $F$ must be the specific correction term intended by the source, rather than an arbitrarily chosen difference of the two series.

Unresolved formulation. The record does not specify a ground field, additional hypotheses on $S$ or $E$, the enumerative invariants used as coefficients, curve classes and their allowed range, generating variables, normalizations, or the formula for $F$. It also does not specify the algebra containing the series, any change of variables or expansion convention, or whether the equality is formal or analytic. Thus the displayed equation preserves the supplied target but is only a schematic statement: a precise mathematical identity cannot be reconstructed from this record alone.

The supplied literature assessment refers to a proved "first equality" and a "second equality" remaining open in full generality, whereas the supplied mathematical statement displays only one equality. Their relationship to the displayed equation is unresolved.

Short English statement

The question asks whether the intended stable-pairs series for $S\times E$ equals the intended Gromov--Witten series plus a prescribed correction term. The record omits the definitions and conventions needed to make this a precise identity. Its partially solved label and literature claims are supplied dated curation, requiring independent verification.

Sources

• Local UnsolvedMath registry: permanent ID $30004470$, source problem code catalogue entry, title Stable-Pairs and Gromov--Witten Series for K3 Times Elliptic Curves, in the Oberwolfach Reports Open Problems collection. The supplied record lists proposed year 2020 and supplies no individual proposer.
• Supplied report citation: Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects (2021), Oberwolfach Reports, pp. 977--1021, DOI 10.4171/owr/2020/19. Original OWR report: https://doi.org/10.4171/owr/2020/19. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled Oberdieck--Pixton: https://arxiv.org/abs/1706.10100.
• Supplied literature source, labeled quasimaps to K3 sheaf moduli: https://arxiv.org/abs/2111.11425.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that the first equality of the stated Igusa cusp form conjecture is proved, that quasimaps establish important wall-crossing and Hilbert-scheme cases, and that the second equality remains open in full generality. This is supplied dated curation derived from the supplied OWR curation, not an independently verified assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.