---
title: "The Campana–Peternell conjecture in dimension six"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 067; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1925
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "The Campana–Peternell conjecture in dimension six"
    url: "https://github.com/openai/math/blob/main/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# The Campana–Peternell conjecture in dimension six

## One-sentence takeaway

OpenAI's result family 067 (Algebraic and complex geometry) claims: Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.

## 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

- *The Campana–Peternell conjecture in dimension six*: We prove that every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous, resolving the Campana–Peternell conjecture in complex dimension six. As a consequence, a connected compact Kähler manifold X with nef holomorphic tangent bundle and $\dim_{\mathbb C}X-\widetilde q(X)\leq6$ has ordinary universal cover $F\times\mathbb C^{\widetilde q(X)}$, where F is a rational homogeneous manifold and $\widetilde q(X)$ is the maximal irregularity of a connected finite étale cover.

## 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: [The Campana–Peternell conjecture in dimension six](https://github.com/openai/math/blob/main/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.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