Numerical semiampleness of projective adjoints
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Statement recorded in the supplied source.
For a projective rational klt pair (X,B) over an algebraically closed characteristic-zero field, assume $K_X+B$ is pseudo-effective and M is a nef rational Cartier divisor. If $D=K_X+B+M$ is nef, must D be numerically equivalent to some semiample rational Cartier divisor?Short English statement
For a projective rational klt pair (X,B) over an algebraically closed characteristic-zero field, assume $K_X+B$ is pseudo-effective and M is a nef rational Cartier divisor. If $D=K_X+B+M$ is nef, must D be numerically equivalent to some semiample rational Cartier divisor?Sources
- S1 Original question source or mathematical context. https://arxiv.org/abs/0907.5182.
- S2 Original question source or mathematical context. https://arxiv.org/abs/1005.2796.
- S3 Original question source or mathematical context. https://arxiv.org/abs/1805.01600.
- S4 Original question source or mathematical context. https://arxiv.org/abs/2406.18233.
- S5 Original question source or mathematical context. https://arxiv.org/abs/2505.05250.
- S6 Original question source or mathematical context. https://arxiv.org/abs/2601.15138.
- S7 Original question source or mathematical context. https://arxiv.org/abs/2604.03879.
- S8 Original question source or mathematical context. https://arxiv.org/abs/2604.28085.
- S9 Original question source or mathematical context. https://doi.org/10.4171/owr/2009/44.
- S10 Original question source or mathematical context. https://ems.press/journals/lem/articles/13750344.
- S11 Original question source or mathematical context. https://www.ams.org/jams/2010-23-02/S0894-0347-09-00649-3/viewer/.
- S12 Original question source or mathematical context. https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/two-nonvanishing-results-concerning-the-anticanonical-bundle/1E68F2B88B818A7CBD47B96B7D700548.
- S13 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2576
Manuscript 1 (family 036, 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.
Numerical semiampleness of projective adjoints
OpenAI's claim
For projective characteristic-zero klt pairs, a nef ordinary adjoint plus a nef rational divisor is claimed to be numerically equivalent to a semiample divisor whenever the ordinary adjoint is pseudo-effective. 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 a rational projective pair with klt singularities $(X,B)$ over a characteristic-zero field assumed algebraically closed, assume $K_X+B$ pseudo-effective and $M$ nef rational Cartier. If $D=K_X+B+M$ is nef, OpenAI claims \[ \exists\text{ semiample rational Cartier }L\text{ on }X:\quad D\equiv L. \] Numerical equivalence is equality of degrees on every integral curve. Semiampleness is claimed for a numerically equivalent replacement, not necessarily for the original actual divisor $D$.Outline of the argument
Manuscript 1 (family 036, 2026-10-03): Ordinary log abundance and terminating programs supply the birational reductions. The remaining full-nef-dimension Calabi--Yau case is treated with jet and Frobenius estimates, after which the positive part descends as a semiample numerical representative; flat nontorsion bundles explain why numerical equivalence is the conclusion.Formal verification
Family 036 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, Numerical Semiampleness of Nef Adjoint Divisors, 2026-10-03; citation key
OAI:Numerical-Semiampleness-of-Nef-Adjoint-Divisors-October-3-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.