A uniform pluricanonical Iitaka degree
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
For each $d\geq 1$, does there exist a positive m(d) such that $|m(d)K_X|$ gives the entire Iitaka fibration of every smooth integral projective d-fold of nonnegative Kodaira dimension over every algebraically closed characteristic-zero field? The ratios must generate the Iitaka field, rather than merely attain its transcendence degree.Short English statement
For each $d\geq 1$, does there exist a positive m(d) such that $|m(d)K_X|$ gives the entire Iitaka fibration of every smooth integral projective d-fold of nonnegative Kodaira dimension over every algebraically closed characteristic-zero field? The ratios must generate the Iitaka field, rather than merely attain its transcendence degree.Sources
- S1 Original question source or mathematical context. https://arxiv.org/abs/1005.2796.
- S2 Original question source or mathematical context. https://arxiv.org/abs/2406.18233.
- S3 Original question source or mathematical context. https://arxiv.org/abs/2505.05250.
- S4 Original question source or mathematical context. https://arxiv.org/abs/2601.15138.
- S5 Original question source or mathematical context. https://arxiv.org/abs/2604.03879.
- S6 Original question source or mathematical context. https://arxiv.org/abs/2604.28085.
- S7 Original question source or mathematical context. https://arxiv.org/html/2501.04232v1.
- S8 Original question source or mathematical context. https://ems.press/journals/lem/articles/13750344.
- S9 Original question source or mathematical context. https://www.ams.org/jams/2010-23-02/S0894-0347-09-00649-3/viewer/.
- S10 Original question source or mathematical context. https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/two-nonvanishing-results-concerning-the-anticanonical-bundle/1E68F2B88B818A7CBD47B96B7D700548.
- S11 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2594
Manuscript 1 (family 034, 2026-10-03): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.
A uniform pluricanonical Iitaka degree
OpenAI's claim
A degree depending only on dimension is claimed to define every characteristic-zero smooth projective Iitaka fibration with nonnegative Kodaira dimension and to recover its whole function field. The cited manuscript argues for that result; its relation to this problem is specified below.Mathematical statement of the claimed result
Mathematical claim 1.
For every $d\ge1$, OpenAI claims a positive integer $m(d)$ valid for every smooth integral projective $d$-fold $X$ over every algebraically closed characteristic-zero field with $\kappa(X)\ge0$, such that \[ |m(d)K_X|\ne\varnothing,\qquad k(s/t:\ 0\ne s,t\in H^0(X,m(d)K_X))=K(K_X). \] The field on the right is generated by all equal-degree positive pluricanonical section ratios. Thus the one complete system defines the Iitaka fibration, not merely a map of the correct image dimension.Outline of the argument
Manuscript 1 (family 034, 2026-10-03): Degeneration of individual generic-fiber factors bounds their volume-form characters independently of semistable ramification. Weak positivity and flattening supply compatible small-volume bounds on the base; Frobenius-diagonal estimates and bounded curve chains then complete the dimension induction.Formal verification
Family 034 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.
Status caveat
OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.Sources
- OpenAI, Uniform Pluricanonical Iitaka Fibrations, 2026-10-03; citation key
OAI:Uniform-Pluricanonical-Iitaka-Fibrations-October-3-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.