Holomorphic Anomaly Equations for Elliptic Threefolds

Definitions and mathematical statement

Let $\pi\colon X\to B$ be the elliptically fibered threefold in the supplied record. Write $T_X$ for its tangent bundle and $\omega_X$ for its canonical line bundle, wherever the intended geometric hypotheses make these bundles defined. Assume the stated vanishing \[ c_3(T_X\otimes\omega_X)=0. \] The record does not specify the ground field, smoothness or properness hypotheses, the precise meaning of elliptic fibration, or the group in which this Chern class vanishes; these remain qualifications requiring recovery from the cited source.

The extracted text uses an effective base curve class $\beta\in H_2(B,{\mathbb{Z}})$, descendant insertions $\operatorname{ch}_{k_i}(\gamma_i)$, and a normalized relative Pandharipande--Thomas descendant series denoted schematically by \[ Z_\beta\!\left(\operatorname{ch}_{k_1}(\gamma_1)\cdots \operatorname{ch}_{k_n}(\gamma_n)\right). \] Here the insertion classes belong to the intended cohomology of $X$. The coefficient ring, allowed indices, relative geometry, curve-class summation, and normalization are not defined by the supplied record. Thus this notation names the source's intended series without supplying a construction of it.

Conditional statement requested by the source. Assuming both the displayed Chern-class vanishing and the preceding conjecture referred to in the extracted Conjecture 3, do these normalized relative Pandharipande--Thomas descendant series satisfy both Oberdieck--Schimpf holomorphic anomaly equations, involving differentiation with respect to $A$ and $G_2$? The additional assumption of the preceding conjecture occurs in the original extract even though it is absent from the cleaned question.

Unresolved formulation. The supplied original text is a corrupted extraction: the equation involving $\partial/\partial G_2$ has no recoverable complete right-hand side, and the ordering and terms of the equation involving $\partial/\partial A$ cannot be certified from the extract. Consequently, the two identities cannot be stated explicitly and reliably from the supplied evidence.

The surrounding extract also contains modified descendants with a $1/24$ correction involving $c_2(T_X)$; a grading by eigenvalues of $[W,\pi^*\pi_*]$; effective decompositions $\beta=\beta_1+\cdots+\beta_\ell$ with divisibilities $\operatorname{div}(\beta_i)$ in $H_2(B,{\mathbb{Z}})$; coefficients $\phi_\alpha\in\mathrm{QJac}$; and theta and discriminant factors, with \[ \Delta(q)=q\prod_{m\geq1}(1-q^m)^{24}. \] It displays the weight and index expressions \[ K_X\cdot\iota_*\beta+ \sum_{i=1}^{n}\left(k_i-1+\operatorname{wt}(\gamma_i)\right), \qquad \frac12\,\beta\cdot\left(\beta+c_1(N_{B/X})\right). \] These are retained as source context, not as independently established constraints: $\iota$, $W$, $N_{B/X}$, the intersection conventions, $\mathrm{QJac}$, the theta normalization, and the anomaly variables are not defined. The occurrence of $N_{B/X}$ and $\iota$ suggests additional geometric data whose existence must not be silently assumed. A complete mathematical formulation requires the intact preceding conjecture and both intact anomaly equations.

Short English statement

The source asks whether normalized relative Pandharipande--Thomas descendant series of an elliptically fibered threefold with $c_3(T_X\otimes\omega_X)=0$ satisfy two Oberdieck--Schimpf holomorphic anomaly equations. Its original extract additionally assumes a preceding conjecture. Missing geometric hypotheses, undefined normalization, and corrupted equations prevent a complete explicit statement from this record alone.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30005167$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection; project registry data/unsolvedmath/problems.json, with supplementary records in catalog/problems.json. These paths identify the local registry context; no file inspection is asserted.
• Supplied source citation: Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects (2023), Oberwolfach Reports, pp. 1805--1864, DOI 10.4171/owr/2022/32. https://doi.org/10.4171/owr/2022/32. The extract labels the target as Conjecture 3, attributed to O.--Schimpf; the cleaned record names Oberdieck--Schimpf. The proposed year is recorded as 2022.
• Supplied literature source: Oberwolfach Report 32/2022. https://doi.org/10.4171/owr/2022/32.
• Supplied literature source: Oberdieck--Schimpf, PT theory of elliptic threefolds. https://arxiv.org/abs/2308.09652.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that elliptic-threefold quasi-Jacobi and anomaly conjectures are proved in special geometries and equivariant or reduced theories, while the general conjectures remain unproved. This is supplied dated curation, not independently verified here; its time-sensitive status and cited claims require independent checking.
• The registry labels its statement corrected_verified and reports that the official report was checked and the notation repaired. This is a supplied verification claim, not a claim of source consultation or mathematical verification by this draft.