View source for Bloch's conjecture for surfaces with geometric genus zero

'''Bloch's conjecture for surfaces with geometric genus zero''' predicts that for a smooth projective complex surface ''S'' with ''p''<sub>''g''</sub>(''S'') = 0, the Albanese map from degree-zero zero-cycles modulo rational equivalence to the Albanese variety is an isomorphism. When also ''q''(''S'') = 0, this says that any two points of ''S'' are rationally equivalent. A manuscript in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 24 September 2026, claims the conjecture for every such surface, with integral coefficients. The family has no Lean formalization. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** Mumford showed that a nonzero holomorphic two-form prevents a finite-dimensional parametrization of zero-cycles, and Bloch conjectured the converse at the level of the Albanese kernel. Bloch, Kas and Lieberman proved the integral assertion for every surface of Kodaira dimension less than two, so the remaining case concerns surfaces of general type with ''p''<sub>''g''</sub> = ''q'' = 0.
** Earlier cases were proved with additional geometry of the surface: the group-action method of Inose and Mizukami and its extension by Barlow, Bauer's treatment of Inoue surfaces with ''K''<sup>2</sup> = 7, Pedrini and Weibel's transcendental-motive arguments for bidouble covers (examples for each 1 ≤ ''K''<sup>2</sup> ≤ 8), and Voisin's theorem on families of surfaces, applied to Catanese and Barlow surfaces.
** The manuscript cites two recent preprints as proposed proofs with overlapping scope: Guletskiĭ ([https://arxiv.org/abs/2512.12451 arXiv:2512.12451]), whose Theorem A covers the full case ''p''<sub>''g''</sub> = ''q'' = 0 over the complex numbers, and Banerjee ([https://arxiv.org/abs/2508.13594 arXiv:2508.13594]), for minimal surfaces of general type with ''p''<sub>''g''</sub> = ''q'' = 0 and ''K''<sup>2</sup> = 9.

== The original result ==
* '''Claim.''' Theorem 1.1 of ''Bloch's conjecture for surfaces with p<sub>g</sub> = 0'' states that for a smooth connected projective complex surface ''S'' with ''p''<sub>''g''</sub>(''S'') = ''q''(''S'') = 0, the degree map from the integral Chow group of zero-cycles to the integers is an isomorphism, with no minimality, fundamental-group or construction hypothesis. Corollary 1.2 combines this with the theorem of Bloch, Kas and Lieberman to obtain Bloch's conjecture for every surface with ''p''<sub>''g''</sub> = 0. The manuscript also states consequences for rational Chow motives of powers and Hilbert schemes of points of such surfaces.
* ''Method, as described:'' after reductions to a minimal surface of general type or a point blowup, the goal is a relation ''c''[Δ] + Γ = 0 in the rational Chow group of ''X'' × ''X'' with ''c'' ≠ 0 and Γ a sum of external products, which kills degree-zero zero-cycles. The relation comes from virtual localization on a Quot scheme of rank-zero quotients of a sum of two line bundles with fixed determinant, built with derived determinant constructions, and a Chow-level version of the Ellingsrud–Göttsche–Lehn recursion that keeps two surface factors unintegrated. The nonvanishing of ''c'' is proved separately.
* '''Source.''' The manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026 Bloch's conjecture for surfaces with p<sub>g</sub> = 0] at commit <code>1581d4c</code>.
* '''Formalization.''' The repository has no Lean scope note for this family.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 20 October 2026)
|-
| 2026-10-16
| J. Jardine, Lattice24, [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_251.json referee review] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Reports no gap. The reviewer states that it read every section except the full details of the derived-geometry section and the conclusion. Its two minor notes: the case ''K''<sup>2</sup> = 9 rests on a comparison whose canonical-divisor multiplier depends on a sign from a local cosection computation and a result of Chang and Kiem, so that a wrong sign would make the key coefficient vanish exactly when the torsion of the Picard group has order two; the paper argues the sign in two independent ways, and the reviewer calls this a point of fragility rather than a gap. The foundational inputs, the derived obstruction theory, virtual pullback and virtual localization, were read but not re-derived.
| Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscript; the two recent preprints are described there as proposed proofs.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]