The effective Iitaka fibration conjecture
For a smooth projective variety with Kodaira dimension , the pluricanonical systems define the Iitaka fibration for all sufficiently large divisible . For varieties of general type, Hacon-McKernan, Takayama and Tsuji found one depending only on ; Birkar and Zhang proved an effective bound that also depends on invariants of the general fibre. Is there, for each , an integer such that defines the Iitaka fibration of every smooth projective -fold with ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Birational geometry; pluricanonical systems
- Posed by
- Stated as Conjecture 1.1 by C. Birkar and D.-Q. Zhang, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs (Publ. Math. IHES, 2016)
- Year posed
- 2016
- Years open
- 10y
- Solved
- 2026-10-03
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Contested
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 32 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for each there is such that, over every algebraically closed field of characteristic zero, is nonempty and its section ratios generate the Iitaka function field for every smooth integral projective -fold with ; every multiple of works too. Theorem 1.2 gives a uniform with for projective lc pairs with and coefficients in a DCC set . The bound is existential; no explicit value is given.
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: Theorem 1.1 of 'Uniform Pluricanonical Iitaka Fibrations' was read against Birkar-Zhang's Conjecture 1.1 as the manuscript describes it; it asks that the system be nonempty and that its section ratios generate the whole Iitaka function field, which is the conjecture's conclusion and slightly stronger than asking for an image of the right dimension. The proof was not refereed. No Lean formalization. The manuscript says it uses the family's log abundance theorem as an input, so this entry depends on that unreviewed claim. 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.