---
title: "Bloch’s conjecture for complex surfaces"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 040; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1900
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Bloch’s conjecture for surfaces with p_g=0"
    url: "https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Bloch’s conjecture for complex surfaces

## One-sentence takeaway

OpenAI's result family 040 (Algebraic and complex geometry) claims: 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.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): This combines the new $p_g=q=0$ theorem with the classical theorem of Bloch, Kas, and Lieberman.
- *Bloch’s conjecture for surfaces with p_g=0*: 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Bloch’s conjecture for surfaces with p_g=0](https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math