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Bloch’s conjecture for surfaces with p_g=0
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Abstract
We prove Bloch's conjecture for smooth connected projective complex surfaces with $p_g=0$: the Albanese homomorphism on integral degree-zero zero-cycles is an isomorphism.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Raemon and commenters (LessWrong), Are OpenAI's math results "creative" in an important way? (LessWrong, post dated 7 Oct 2026). A comment by user Archimedes reports that Simon Pepin Lehalleur, on what is described as "a very superficial reading", judged this proof "not based on deep new insights into codimension 2 cycles in general" but "very substantial progress on cycles on (powers of) surfaces"; the underlying X post is linked from the comment.
Related prior literature mentioned by others
- V. Guletskii, The Bloch conjecture (arXiv:2512.12451, 13 Dec 2025). Earlier preprint claiming the same scope (Albanese kernel vanishes when $H^2$ is algebraic); the family-040 card in Xie's arXiv:2610.17090 lists it as a competing unrefereed proposed proof, and it is cited in the paper.
- K. Banerjee, Bloch's conjecture on surfaces with $p_g=q=0$, $K^2=9$ (arXiv:2508.13594, v1 19 Aug 2025, v2 19 Sep 2026). Proposed proof for the $K^2=9$ case, which the paper treats by a separate branch; also listed in Xie's card as a competing unrefereed proposed proof and cited in the paper.
- Davesh Maulik and Andrei Neguţ, Lehn's formula in Chow and Conjectures of Beauville and Voisin (arXiv:1904.05262, 10 April 2019). Named in Xie's survey (§9.1.4) as related work on Chow universality; it extends Lehn's formula and the Li–Qin–Wang algebra action to Chow groups of Hilbert schemes of arbitrary smooth projective surfaces.
Possibly relevant literature (found by the agent, not checked)
- S. Stark, Cosection localization and the Quot scheme $\mathrm{Quot}^l_S(E)$ (arXiv:2107.08025, Proc. R. Soc. A 478 (2022)). Uses cosection localization along a smooth canonical curve to compute virtual classes of surface Quot schemes, which may bear on the paper's Chang–Kiem/Kiem–Li cosection reduction to fibers over the empty and canonical divisors.
- W. Lim, Virtual $\chi_{-y}$-genera of Quot schemes on surfaces (arXiv:2003.04429, 2020–2023). Expresses Quot-scheme virtual invariants on surfaces through Seiberg–Witten invariants, including minimal surfaces of general type, which may be related to the paper's fixed-locus contributions indexed by effective divisor pairs.
- Andrei Neguţ, The Chow of $S^{[n]}$ and the universal subscheme (arXiv:1912.03287, 6 December 2019). Its description of Hilbert-scheme Chow classes as pushforwards of universal-subscheme Chern-class polynomials with coefficients from powers of the surface may provide context for the paper's claimed Chow-valued universality with two surface factors retained.
- K. Banerjee, Bloch's conjecture for fake projective planes with an automorphism of order 3 (arXiv:2504.08303, v1 11 Apr 2025, v2 12 Sep 2026). Earlier special-case result via involutions and quotient surfaces for some $p_g=q=0$ surfaces and certain fake projective planes, relevant to the paper's survey of group-action approaches and its $K^2=9$ branch.
Digests and expositions
- Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics (arXiv:2610.17090, 12 Oct 2026). Section 9.1.4 (status "not formalized | new strategy") identifies the marked Chow relation from Quot-scheme localization and the universal diagonal coefficient as the new instruments, discusses Chow universality and the Cartwright–Steger comparison, and notes caveats about obstruction-theoretic foundations and that the integral statement does not give an integral Tate-motive decomposition. [AI-generated per source] [family-level]
- AI Proved What?, All points alike? Bloch's conjecture for complex surfaces (episode 040, October 2026). Video and transcript explaining zero-cycles, the claimed scope, the diagonal-relation strategy and the two numerical branches of the proof. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 040
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026,
author = {{OpenAI}},
title = {{Bloch's conjecture for surfaces with $p_g=0$}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf}{OAI:Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026}},
year = {2026}
}Family 040: Bloch’s conjecture for complex surfaces
Proves Bloch's conjecture: for every smooth connected projective complex surface S with $p_g(S)=0$, the Albanese map $\mathrm{CH}_0(S)^0\to\mathrm{Alb}(S)(\mathbb C)$ on integral degree-zero zero-cycles is an isomorphism. This combines the new $p_g=q=0$ theorem with the classical theorem of Bloch, Kas, and Lieberman.
This is the only paper in this family.