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Bloch's Conjecture for Surfaces with pg=0p_g=0

For a smooth connected projective complex surface SS, the Albanese map CH0(S)0→Alb(S)(C)\mathrm{CH}_0(S)^0\to\mathrm{Alb}(S)(\mathbb{C}) on degree-zero zero-cycles modulo rational equivalence is surjective. Mumford showed that pg(S)>0p_g(S)>0 makes CH0(S)\mathrm{CH}_0(S) infinite-dimensional, and Bloch conjectured the converse. Bloch-Kas-Lieberman proved it for Kodaira dimension below 2, leaving surfaces of general type with pg=q=0p_g=q=0, where it says any two points are rationally equivalent; only special families (Godeaux, Campedelli, Burniat, Inoue, Barlow, Catanese) were known. If pg(S)=0p_g(S)=0, is the Albanese map an isomorphism?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Algebraic geometry; algebraic cycles
Posed by
Spencer Bloch
Year posed
1976
Years open
50y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
55 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For every smooth connected projective complex surface SS with pg=q=0p_g=q=0, deg⁡:CH0(S)→Z\deg:\mathrm{CH}_0(S)\to\mathbb{Z} is an isomorphism with integral coefficients, with no minimality, fundamental-group or construction hypothesis. With Bloch-Kas-Lieberman this gives CH0(S)0≅Alb(S)(C)\mathrm{CH}_0(S)^0\cong\mathrm{Alb}(S)(\mathbb{C}) for every such surface with pg=0p_g=0. Over C\mathbb{C} only; it does not address the generalized Bloch conjecture in higher dimension. An earlier proof of the same statement was proposed by Guletskii (arXiv:2512.12451, December 2025), which the manuscript cites; this entry claims no priority over it.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the main theorem and corollary were read against Bloch's conjecture as the manuscript states it. The proof, via virtual localization on a Quot scheme producing a diagonal relation in CH^2(X x X), was not refereed. No Lean formalization is listed. The paper notes two earlier proposed proofs by humans: Guletskii (arXiv 2512.12451, December 2025) claiming the same scope, and Banerjee (arXiv 2508.13594) for K^2 = 9.

Sources

Changelog1 change

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