The Campana–Peternell conjecture
From OpenAI Aftermath
The Campana–Peternell conjecture asserts that every smooth complex projective Fano variety whose tangent bundle is nef is rational homogeneous, that is, isomorphic to G/P for a connected semisimple complex algebraic group G and a parabolic subgroup P. It is a nef analogue of Mori's characterization of projective space by ampleness of the tangent bundle. A manuscript dated 25 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims the conjecture in dimension six. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; this page does not certify them.
Background
- As recounted in the introduction of the manuscript:
- Campana and Peternell proved the conjecture in dimensions at most three and for fourfolds of Picard number greater than one. The remaining fourfold case was settled through Mok's and Hwang's study of the tangent directions of minimal rational curves. Watanabe proved the fivefold case of Picard number greater than one, and Kanemitsu completed the fivefold classification.
- Kanemitsu also proved that a Fano n-fold with nef tangent bundle is rational homogeneous whenever its Picard number exceeds n − 5, which for sixfolds leaves only Picard number one. In that range standard characterization results reduce the problem to pseudoindices four and five, and Watanabe treated the pseudoindex-four sixfold case.
The original result
- Claim. Theorem thm:main of The Campana–Peternell conjecture in dimension six states that every smooth connected complex projective Fano variety of dimension six with nef tangent bundle is rational homogeneous. As a consequence, the manuscript states that a connected compact Kähler manifold with nef tangent bundle whose dimension exceeds its maximal irregularity by at most six has universal cover a product of a rational homogeneous manifold and a complex affine space.
- Method, as described: the manuscript gives an independent intersection argument for pseudoindex four. In the principal new case, pseudoindex five, characteristic-class identities obtained from the universal family of minimal rational curves, viewed both as a family of curves and as a family of threefolds over the variety, and an exact finite elimination determine the variety of minimal rational tangents. Mok's recognition theorem then identifies the Grassmannian of planes in a five-dimensional space. A script in the manuscript's verification folder carries out the exact arithmetic certificate.
- Sources. At commit
1581d4c: The Campana–Peternell conjecture in dimension six. - Formalization. The repository's Lean documentation has no scope note for this family at that commit.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 24 October 2026) |
|---|---|---|---|---|
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_276 of the manuscript in JJardine919/openai-math-audit | Review (AI-assisted audit) | Reports no gap and no minor issue after reading every section. It checks by hand the pseudoindex-four argument, the moment and characteristic-class identities of the pseudoindex-five case and the sectional-genus exclusion, reruns the manuscript's verification script, and adds an independent check by torus localization on a flag variety for the Grassmannian. | Part of an audit of the collection's manuscripts that the audit describes as complete. |
| 2026-10-23 | GitHub account maximusrufus, maximusrufus/openai-math-replays, replay e067_campana_peternell at commit 4c5524a |
Verification (computational replay) | Replays the finite computations stated in the manuscript from its text, with exact arithmetic; 33 of 33 replayed claims pass. It states that the geometric derivation of the constraints is not replayed. | GitHub repository with continuous-integration reruns. It states that a replay shows that the stated computations are consistent with the manuscript, not that the theorem is true. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
