Fujita's freeness conjecture
Let be a smooth complex projective variety of dimension , its canonical bundle and an ample line bundle. Fujita conjectured that is globally generated for every (and very ample for ); the bound is sharp for . Known: curves by Riemann-Roch, surfaces by Reider, dimensions 3, 4, 5 by Ein-Lazarsfeld, Kawamata and Ye-Zhu; in general Angehrn-Siu's quadratic bound , later improved to by Ghidelli-Lacini and to about by Han. Is globally generated for every smooth projective of every dimension and every ample ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Algebraic geometry; adjoint linear systems
- Posed by
- Takao Fujita, On polarized manifolds whose adjoint bundles are not semipositive, Algebraic Geometry, Sendai 1985, Adv. Stud. Pure Math. 10 (1987)
- Year posed
- 1987
- Years open
- 39y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 60 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every smooth connected complex projective variety of dimension and every ample line bundle , is generated by global sections; hence is globally generated for all , even when itself is not. The bound is sharp. The method minimises an exponential average of normalised section orders over bases and weighted monomial valuations, then lifts sections with Fujita's discrepancy-and-vanishing mechanism. It does not prove the very-ampleness conjecture () or point separation.
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 README also cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against Fujita's conjecture as the manuscript cites it. The proof was not refereed. No Lean formalization: the manuscript is not in lean/formalization.yaml and lean/docs/038.md does not exist at the pinned commit. Scope the paper itself states: only the freeness (global generation) part is proved, not separation of points or tangents and not the very-ampleness half of Fujita's conjecture. Readers should know a previous claimed proof (Chan, arXiv 2411.07129) was withdrawn in November 2024; the manuscript cites that withdrawal. A brief web search on 7 October 2026 found no public dispute of this manuscript.