---
id: "2026-10-06-openai-bloch-conjecture-surfaces-claim"
url: "https://postcutoff.com/e/2026-10-06-openai-bloch-conjecture-surfaces-claim/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Expert-checked
sources: 4
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-bloch-conjecture-surfaces-claim/

# OpenAI claims Bloch's conjecture for surfaces

Full title: OpenAI's release claims Bloch's conjecture on zero-cycles for every complex surface with p_g = 0

"Bloch's conjecture for surfaces with p_g = 0" (OpenAI, 24 Sep 2026, family 040 of the 6 Oct release) proves that for a smooth connected projective complex surface S with p_g(S) = q(S) = 0 the degree map CH₀(S) → ℤ is an isomorphism, with integral coefficients and no further hypotheses; combined with Bloch–Kas–Lieberman this gives Bloch's conjecture that the Albanese map on degree-zero zero-cycles is an isomorphism whenever p_g = 0. Earlier proofs covered specific general-type surfaces (Godeaux, Campedelli, Barlow, Catanese and others). The argument obtains a Chow relation from virtual localization on a Quot scheme. It is unrefereed and has no Lean formalization.

## Key facts
- Theorem 1.1 (verbatim): 'Let S be a smooth connected projective surface over C with pg(S) = q(S) = 0. Then deg : CH0(S) → Z is an isomorphism, where CH0(S) has integral coefficients.' The paper: 'Theorem 1.1 imposes no minimality, fundamental-group, or surface-construction hypothesis.' Corollary 1.2 is Bloch's conjecture for p_g = 0. 45 pages, dated 24 Sep 2026
- Prior state (paper): Mumford showed a nonzero holomorphic 2-form makes zero-cycles infinite-dimensional; Bloch, Kas and Lieberman proved the conjecture for Kodaira dimension < 2; general-type cases were done family by family (Inose–Mizukami, Barlow, Bauer, Pedrini–Weibel, Voisin for Catanese and Barlow surfaces)
- Earlier proposals: the paper cites Guletskiĭ's preprint (arXiv 2512.12451, Dec 2025), whose Theorem A covers the same scope, and Banerjee's proof for minimal surfaces with K² = 9, 'as proposed proofs'. Its own route: 'the argument developed here obtains its Chow relation from virtual localization on a Quot scheme', using derived Quot schemes and Marian–Oprea–Pandharipande-type obstruction theories
- Stated consequence: for p_g = q = 0, every rational Chow class on a power of S or on a Hilbert scheme of points is determined by its cohomology class (finite-dimensional, pure Tate motives in Kimura's sense)
- Daniel Litt (X, 6 Oct, verbatim): 'Proof of Bloch's conjecture! I thought about that a bit in grad school. Paper seems to cite some wrong existing work without mentioning it's wrong :-/ but AFAICT doesn't meaningfully rely on it.' (He did not name the work)
- Kiwamu Watanabe listed the release's algebraic-geometry claims as 'quite amazing' but said he could not judge them (see [2026-10-06-openai-math-release-722-manuscripts](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)). No other expert assessment found by 29 Oct

## What happened

Zero-cycles on a surface are formal sums of points, and two cycles are rationally equivalent when they differ by divisors of rational
functions on curves. Mumford showed in 1969 that a surface with a nonzero holomorphic 2-form has an enormous, infinite-dimensional
group of zero-cycles. Spencer Bloch conjectured the converse: if p_g = 0, then a zero-cycle is determined by its degree and its image
in the Albanese variety. Bloch, Kas and Lieberman proved this for surfaces that are not of general type, leaving the general-type
surfaces with p_g = q = 0, for which the prediction is that any two points are rationally equivalent. Since then the conjecture has
been checked only for particular families such as Godeaux, Campedelli, Barlow and Catanese surfaces.

OpenAI's 45-page manuscript claims the general-type case with no assumptions on how the surface is built. Instead of using symmetries
or explicit families, it produces the needed relation among zero-cycles by virtual localization on a Quot scheme of the surface, using
derived moduli and enumerative-geometry techniques. It also states consequences for Chow motives of powers and Hilbert schemes of S.

## Why it matters

Bloch's conjecture is one of the central open problems about algebraic cycles, a test case of the Bloch–Beilinson conjectures. A
uniform proof would replace fifty years of case-by-case work, but at present it rests on an unrefereed, unformalized manuscript.

## What is disputed or not yet verified

- Verification: Unrefereed manuscript; no Lean formalization; no published expert check

## Reactions

Daniel Litt, who had worked on the problem as a graduate student, welcomed the claim but noted that the paper cites some earlier work
he considers wrong without saying so, adding that it does not seem to rely on it. The manuscript itself lists a 2025 proof proposal by
Guletskiĭ covering the same statement as "proposed". No specialist has published a reading of the argument.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: Bloch's conjecture for surfaces with p_g = 0 (24 Sep)](https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf) (github.com, paper)
2. [arXiv 2512.12451: V. Guletskii, The Bloch conjecture (Dec 2025, cited as a proposed proof)](https://arxiv.org/abs/2512.12451) (arxiv.org, paper)
3. [Daniel Litt on X: 'Proof of Bloch's conjecture!' (6 Oct)](https://x.com/littmath/status/2107616410897678765) (x.com, discussion)
4. [oai-math-revised: community notes on the Bloch manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29 (filed): Created (families 001–125; manuscript introduction and Litt's post via fxtwitter read)

## Related

- 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)
- People: [Daniel Litt](https://postcutoff.com/person/daniel-litt/)