The Campana-Peternell conjecture on Kodaira dimension
Campana and Peternell, studying positivity of the cotangent bundle, conjectured that if on a smooth projective variety , with , effective and pseudo-effective, then . The case is canonical nonvanishing. Schnell reduced the conjecture, given nonvanishing, to fibrations whose fibres have Kodaira dimension zero. Is it true that for every smooth connected projective complex variety , effective Cartier divisor and integer with pseudo-effective, ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Birational geometry; Kodaira dimension
- Posed by
- F. Campana and T. Peternell, Geometric stability of the cotangent bundle and the universal cover of a projective manifold (Bull. SMF, 2011), Conjecture 2.4
- Year posed
- 2011
- Years open
- 15y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Contested
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Corollary 1.2: if is pseudo-effective for an effective Cartier (or, after clearing denominators, rational) divisor on a smooth connected projective complex variety , then . Corollary 1.3: if is a fibre space and is pseudo-effective with ample, then and has sections for large (Schnell's general fibre-space conclusion). Both use canonical nonvanishing in every dimension from the family's abundance manuscript. The zero-Kodaira case of Schnell's question was already proved by Zou (2025) and is not new here.
What the AI did
The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. Family 034 has fourteen manuscripts dated September 24 to October 5, 2026, which cite one another as inputs; this entry draws on the manuscript named as its source. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.
Verification
No independent mathematician has checked this yet. Checked here: Corollary 1.2 of 'Schnell fiber spaces and good canonical models' was read against the conjecture as the manuscript quotes it (Campana-Peternell, Conjecture 2.4) in its Kodaira-dimension form. The paper's own new step is a mixed-intersection argument from a good minimal model of the total space; the good models and canonical nonvanishing come from the family's log abundance manuscript, so this entry stands or falls with that one. The zero-Kodaira fibre case (its Theorem 1.1) was already proved by Zou (2025), as the paper says. The proof was not refereed. No Lean formalization. Listed as Contested because it depends on the log abundance entry, which the release itself is inconsistent about; see the claim issue.
Claim issue
This result uses the release's log abundance theorem as an input, and the release contradicts itself on whether that theorem is unconditional (24 September abstract) or a conditional resolution resting on logarithmic Iitaka subadditivity (5 October appendix). The entry is Contested with it.