The log abundance conjecture in characteristic zero (rational boundary, every dimension), with canonical nonvanishing
For a projective log canonical pair the abundance conjecture of the minimal model program asserts that if the log canonical divisor is nef then it is semiample: some positive multiple is generated by global sections, so numerical positivity defines a morphism. The companion nonvanishing conjecture asserts that a pseudo-effective canonical class on a smooth projective variety has a nonzero pluricanonical section. Abundance was known in dimension at most three (Miyaoka, Kawamata, Keel-Matsuki-McKernan) and in special cases beyond. Is every nef -Cartier on a projective log canonical pair with effective rational boundary, over an algebraically closed field of characteristic zero, semiample?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Birational geometry; minimal model program
- Posed by
- Central conjecture of the minimal model program (Kawamata, Miyaoka, Reid and others); the manuscripts name no single poser and cite its threefold history
- Year posed
- —
- Years open
- —
- 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
- 75 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for a normal projective lc pair over an algebraically closed field of characteristic zero, with effective rational and -Cartier and nef, some positive Cartier multiple of is globally generated. The induction also gives good minimal models for lc pairs with pseudo-effective real adjoint and, over , canonical nonvanishing for smooth projective varieties in every dimension; consequences include rational slc abundance (Fujino-Gongyo) and finite generation of lc rings. Not shown: semiampleness with real boundaries, positive characteristic, or any bound on the multiple. The family's compact Kahler log abundance theorem assumes logarithmic Iitaka subadditivity (claimed in family 033) and is not part of this entry.
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 and the introduction of 'Log abundance in characteristic zero' were read against the conjecture as stated there; the claim is unconditional and covers every dimension. The proof (a dimension induction using a signed-boundary criterion, Hodge-module lifting through infinitesimal neighbourhoods and Hashizume's reduction to smooth nonvanishing) was not refereed. No Lean formalization. A reader should know that the release is not consistent about whether projective abundance is unconditional: the October 5 companion on Kahler fourfolds proves projective log abundance in its appendices and calls it a conditional resolution, not an unconditional abundance theorem, assuming logarithmic Iitaka subadditivity, while this September 24 manuscript claims it outright. The family's effective Iitaka, index and Campana-Peternell results use this manuscript as an input. Listed as Contested because the release itself disagrees on whether the theorem is conditional; see the claim issue.
Claim issue
The release contradicts itself on this claim. The 24 September manuscript's abstract states that the rational-boundary log abundance conjecture is proved in every dimension. An appendix of the release's own 5 October manuscript on compact Kahler fourfolds calls the same theorem "a positive conditional resolution of the log abundance conjecture. It is not an unconditional abundance theorem", resting on a logarithmic Iitaka subadditivity assumption that it says follows from a lemma elsewhere in the release. Until OpenAI clarifies which reading is meant, or a specialist checks the chain, the entry is Contested.
Sources
- PaperKahler version (conditional on log Iitaka subadditivity): Log abundance for compact Kähler spaces under logarithmic IitaFourfold nonvanishing by minimal metrics and moving jetsLifting sections from the reduced support of an adjointMinimal metrics and interior injectivity for nef adjointsAbundance after nonvanishing for compact Kähler fourfoldsConditional good minimal models for compact Kähler fourfolds
- CodeOpenAI math release: Log abundance in characteristic zero