Fujita's freeness conjecture
From OpenAI Aftermath
Fujita's freeness conjecture predicts that for a smooth complex projective variety X of dimension n and an ample line bundle L, the adjoint bundle KX + mL is globally generated for every integer m ≥ n + 1. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims a proof in every dimension. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- For curves Riemann–Roch gives the sharp bound, and the surface case follows from Reider's theorem. The threefold, fourfold and fivefold cases were proved by Ein–Lazarsfeld, Kawamata and Ye–Zhu.
- In arbitrary dimension, Angehrn and Siu obtained generation for m ≥ (n2 + n + 2)/2. Helmke developed multiplicity estimates for the centers in the vanishing argument; Heier combined them with the Angehrn–Siu method to reach a bound of order n4/3, and Ghidelli and Lacini obtained m ≥ n(log log n + 2.34) for n ≥ 2.
- A recent preprint of Han gives a linear bound, m ≥ ⌈C0n⌉ with C0 = 1.77629988…; see arXiv:2609.01574, A linear bound for Fujita's freeness conjecture by J. Han, submitted 1 September 2026 and revised 11 September 2026. Chan announced a proof of a stronger intersection-number formulation in a preprint that was later withdrawn.
The original result
- Claim. Theorem 1.1 of Fujita's freeness conjecture states that for a smooth connected projective variety X over the complex numbers of dimension n ≥ 1 and an ample line bundle L, the bundle KX + (n + 1)L is generated by its global sections; a corollary extends this to every m ≥ n + 1. The manuscript notes that the exponent is sharp on projective space and that the very-ampleness part of Fujita's conjecture is not addressed.
- Method, as described: assuming generation fails at a point, the manuscript minimizes an exponential average of normalized section orders over bases of the section space and weighted monomial valuations centered at the point. A first variation in directions tangent to a positive-dimensional center, combined with a uniform restriction estimate and the Brunn–Minkowski inequality, forces the minimizing center to be the point itself; a perturbed ideal and Kawamata–Viehweg vanishing then lift a nonzero section value.
- Sources. The manuscript Fujita's freeness conjecture at commit
1581d4c. - Formalization, as described by the repository. The collection has no Lean scope note or Comparator statement for this result.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 22 October 2026) |
|---|---|---|---|---|
| 2026-10-09 | K. Kitamura, Mixed Fujita Freeness and Linear Effective Base-Point Freeness (KitaKen1/mixed-fujita-freeness, GitHub, commit 0abf53b; PDF labelled public draft 2, 9 October 2026) |
Related (stronger statement) | Proposes proofs that on a smooth projective n-fold the divisor KX + L1 + … + Ls + N is globally generated for any ample Cartier divisors L1, …, Ls with s ≥ n + 1 and any nef Cartier divisor N, which contains the sharp freeness bound as the case of equal Li and N = 0, and that for a projective klt pair a nef Cartier D with aD − KX − Δ nef and big has mD globally generated for every m ≥ a + ν(D) + 1. | Public draft PDF in a GitHub repository, described by its author as proposed proofs; it has no reference list and does not cite the OpenAI manuscript. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_532 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap. The reviewer read the whole paper and checked its computations by hand, but did not re-verify the cited analytic semicontinuity theorem of Demailly–Kollár or Fujino's form of Kawamata–Viehweg vanishing. Its one minor note says that a perturbed monomial test in the center argument is not literally of the admissible type defined in the paper and needs a standard approximation by divisorial valuations, and that the stated discrepancy value is correct. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
