The Campana-Peternell conjecture on Fano manifolds with nef tangent bundle, in dimension six
A smooth complex projective Fano manifold has nef tangent bundle whenever it is rational homogeneous . Campana and Peternell (1991) conjectured the converse: every Fano manifold with nef tangent bundle is rational homogeneous. It was proved in dimension at most three and for fourfolds of Picard number greater than one by Campana-Peternell, for all fourfolds by Mok and Hwang, and for fivefolds by Watanabe and Kanemitsu; Kanemitsu proved it when , which for sixfolds leaves Picard number one, and Watanabe treated the pseudoindex-four sixfolds. Is every smooth complex Fano sixfold with nef tangent bundle rational homogeneous?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Algebraic geometry; Fano manifolds and rational homogeneous spaces
- Posed by
- Frederic Campana and Thomas Peternell
- Year posed
- 1991
- Years open
- 35y
- Solved
- 2026-09-25
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 28 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous. The new step is Picard number one and pseudoindex five, where a characteristic-number identity forces the VMRT to be the Segre threefold , so . Corollary 1.2: a compact Kahler manifold with nef tangent bundle and has universal cover with rational homogeneous. Dimensions seven and higher remain open.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 25, 2026) is the whole family; its folder includes a verification script for the exact arithmetic certificate in Appendix A.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the Campana-Peternell conjecture; it is exactly the conjecture in complex dimension six. The proof was not refereed. It relies on Kanemitsu's theorem for Picard number at least two, Watanabe's pseudoindex-four result (reproved independently), Hwang-Mok VMRT methods and Mok's recognition of the Grassmannian Gr(2,5). No Lean formalization exists for this family (lean/docs/067.md is absent at the pinned commit). The verification/verify.py arithmetic certificate was not run here.