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The effective log Iitaka conjecture for log canonical pairs with finite rational coefficient sets

Chen, Han and Liu conjectured a logarithmic version of effective Iitaka fibrations: for fixed dimension dd and a DCC set Φ⊂[0,1]\Phi\subset[0,1], one integer mm should make the rounded system ∣⌊m(KX+B)⌋∣|\lfloor m(K_X+B)\rfloor| nonempty with section ratios generating the Iitaka field, for every projective lc pair (X,B)(X,B) of dimension dd with coefficients in Φ\Phi and nonnegative Kodaira dimension. They proved it in dimension at most three. Does it hold in every dimension when Φ\Phi is a finite set of rationals?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Birational geometry; pluricanonical systems
Posed by
G. Chen, J. Han and J. Liu, On effective log Iitaka fibrations and existence of complements (IMRN 2024; arXiv 2301.04813), Conjecture 1.2
Year posed
2023
Years open
3y
Solved
2026-09-26
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Contested
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

'Uniform effective log Iitaka fibrations for fourfolds' (September 26): for normal projective lc complex fourfolds with coefficients in a fixed finite rational set and pseudo-effective rational Cartier adjoint, one degree makes the rounded system nonempty with section ratios generating the Iitaka field; it also bounds the canonical index of klt Calabi-Yau fourfolds. 'Uniform log Iitaka fibrations and bounded moduli denominators' (October 4): the same conclusion for d≥5d\ge5 over any algebraically closed field of characteristic zero, with bounded Cartier denominators for moduli divisors. Not shown: the DCC case of the conjecture in dimension four or more.

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: the main theorems of the two manuscripts were read against Conjecture 1.2 as both describe it. Both papers say their coefficient scope (finite rational sets) is narrower than the DCC conjecture, hence partial. The proofs were not refereed. No Lean formalization. Both rely on the family's abundance, nonvanishing and index results, all unreviewed. 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.

Sources

Changelog1 change

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