The generalised abundance conjecture
From OpenAI Aftermath
The generalised abundance conjecture of Lazić and Peternell asserts that for a projective klt pair (X, B) with KX + B pseudo-effective and a nef divisor M, if KX + B + M is nef then it is numerically equivalent to a semiample divisor. Numerical equivalence cannot be replaced by linear equivalence: a non-torsion line bundle of degree zero on an elliptic curve is nef but has no sections in any positive power. For M = 0 the statement is equivalent to the abundance conjecture for klt pairs. Four manuscripts in OpenAI's public mathematics collection (github.com/openai/math), listed together as its result 036 and dated from 24 September to 6 October 2026, claim the conjecture over algebraically closed fields of characteristic zero, a compact Kähler analogue, minimal models or Mori fibre spaces for generalized log canonical pairs, and minimal models in numerical dimension one. The proofs use other manuscripts of the collection, in particular its log abundance theorem (see The log abundance conjecture in characteristic zero). The collection contains no Lean formalization listed for these results.
Background
- As recounted in the introductions of the manuscripts:
- Lazić and Peternell formulated the Generalised Abundance Conjecture, for a nef Cartier summand. Fukuda's theorem shows that a semiample numerical representative makes KX + B itself semiample, so the case M = 0 is ordinary klt abundance.
- The generalized-pair formalism, with Birkar and Zhang's definition and effective birationality, keeps a nef divisor on a higher birational model as part of the adjoint data and accommodates the moduli part of the canonical bundle formula.
- In the compact Kähler setting, Höring and Peternell constructed minimal models for Kähler threefolds, and the threefold abundance results of Campana, Höring and Peternell and the log abundance theorem of Das and Ou are recalled as background.
The original results
- Claim, generalised abundance. Theorem 1.1 of Numerical semiampleness of nef adjoint divisors states that for a projective klt Q-pair (X, B) over an algebraically closed field of characteristic zero with KX + B pseudo-effective and a nef Q-Cartier divisor M on X, if D = KX + B + M is nef then D is numerically equivalent to a semiample Q-Cartier divisor. Its Corollary 1.2 states that on a projective klt pair with KX + B numerically trivial every nef Q-Cartier divisor is numerically semiample. The manuscript states that the proof uses the collection's log abundance theorem and terminating minimal model programs from a companion manuscript.
- Claim, Kähler analogue. Numerical semiampleness of nef adjoint classes on compact Kähler manifolds states that for a compact Kähler manifold with an effective rational simple normal crossing boundary with coefficients less than one and a nef rational line bundle M, if KX + B is pseudo-effective and KX + B + M is nef, then its first Chern class in real Bott–Chern cohomology is represented by a semiample rational line bundle. The abstract states that this allows a flat change of line bundle and does not assert semiampleness of the original adjoint.
- Claim, minimal models. Theorem 1.1 of Minimal models and Mori fibre spaces for generalized log canonical Q-pairs states that a projective generalized log canonical Q-pair over an algebraically closed field of characteristic zero admits a minimal model when its adjoint divisor is pseudo-effective and a Mori fibre space otherwise, with the nef b-divisor fixed; Corollary 1.2 gives the case of ordinary log canonical pairs.
- Claim, numerical dimension one. Theorem 1.1 of Minimal models in numerical dimension one states that a smooth connected complex projective variety of dimension at least three with KX pseudo-effective and numerical dimension κσ(X, KX) = 1 has a projective Q-factorial terminal minimal model; Corollary 1.2 states that this model is a good minimal model.
- Formalization. The collection's catalogue lists no Lean formalization for this result.
- Sources (openai/math repository, commit
7ef5c0c): Numerical semiampleness of nef adjoint divisors, dated 3 October 2026; Numerical semiampleness of nef adjoint classes on compact Kähler manifolds, dated 6 October 2026; Minimal models and Mori fibre spaces for generalized log canonical Q-pairs and Minimal models in numerical dimension one, both dated 24 September 2026.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026; Bintz row 31 October 2026) |
|---|---|---|---|---|
| 2026-10-12 | AI Proved What?, episode 036 Semiample, numerically?, remade version of 26 October 2026 | Reading (exposition) | Video explainer of the claims: Lazić and Peternell's generalized abundance conjecture for projective klt pairs in characteristic zero, that K plus B plus M is numerically equivalent to a semiample divisor when K plus B is pseudo-effective, M is a nef Q-Cartier divisor on X and K plus B plus M is nef, so that every nef divisor on a klt Calabi–Yau pair is numerically semiample; a Kähler version with Bott–Chern classes; minimal models or Mori fibre spaces for generalized log canonical pairs; and minimal models in numerical dimension one. The episode uses an elliptic curve to show why the conclusion is numerical, and states that the nef summand must be a divisor on X and that the minimal-model paper gives a terminating choice of program, not termination of every program. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as preprints: OpenAI, four papers dated September 24 to October 6, 2026, released October 6; under expert review: not yet; peer reviewed: not yet; checked in Lean: no Lean proof of the main theorems. The page states that none of the papers has been withdrawn and that the Kähler paper was revised in the October 7 update to state its inputs explicitly. It states that the central paper depends on the collection's log abundance result and on the generalized minimal-model paper, and the Kähler paper also on the central paper and on the collection's Iitaka subadditivity result, and that it found no public assessment by specialists. It states that an uncatalogued supporting Lean library for the numerical-dimension-one paper, 51 files and 29,042 lines with no unproved steps by its search, proves some ingredients, and that the site has not run it. |
| 2026-10-12 | M. Bintz, thread on recurring techniques in the release, third pair | Reading, comparison of techniques | A thread on techniques that recur across manuscripts of the collection without cross-citation, with the equality sign used loosely. Its third pair matches the manuscript Minimal models and Mori fibre spaces for generalized log canonical Q-pairs with the collection's manuscript Bounded klt complements for Fano contractions: Lemma 8.3 of the first with Proposition 14.3 of the second, and broadly Lemma 6.3 to Section 8.3 of the first with Lemma 11.6 to Proposition 14.5 of the second. It suggests that the shared ideas may come from the proof systems sharing a workspace. | Social media thread on Bluesky. Read on 31 October 2026. |
| 2026-10-16 | J. Jardine, Lattice24, referee reviews rank_329 (generalised abundance), rank_336 (Kähler analogue), rank_321 (generalized log canonical minimal models) and rank_061 (numerical dimension one) in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | All four reviews report no gap and list the steps they checked. Each records as a minor issue that the result rests on large external inputs from companion manuscripts: the review of the generalised abundance manuscript names the collection's log abundance theorem and terminating minimal model programs, both described as major open conjectures that the manuscript itself does not prove, and five lemmas imported from the log abundance manuscript with proof sketches only; the review of the Kähler manuscript names Kähler log abundance in every dimension and other companion results; the review of the minimal model manuscript states that everything rests on a local phase theorem and uniform lifting and bounded-fibre theorems that it did not verify line by line; and the review of the numerical-dimension-one manuscript states that the good-minimal-model corollary relies on the companion log abundance theorem, while the main theorem does not. | Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0. |
Notes
- The background summarizes the manuscripts' own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
