The LeBrun–Salamon conjecture
The LeBrun–Salamon conjecture asserts that every compact positive quaternionic-Kähler manifold, that is, a Riemannian 4m-manifold (m ≥ 2) with holonomy in Sp(m)Sp(1) and positive scalar curvature, is a symmetric Wolf space. Through the twistor construction it is tied to the algebraic conjecture that every contact Fano manifold is homogeneous, namely an adjoint variety. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 23 September 2026, claims both, together with a classification of projective contact manifolds. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Wolf related compact quaternionic symmetric spaces to homogeneous complex contact manifolds. The twistor construction of Salamon, independently of Bérard-Bergery, attaches a contact Fano manifold to a positive quaternionic-Kähler manifold, and LeBrun characterized the contact Fano manifolds arising this way by the existence of a Kähler–Einstein metric. The algebraic formulation is stated, for example, as Buczyński's Conjecture 1.1.
- Kebekus, Peternell, Sommese and Wiśniewski, together with Demailly's integrability theorem, reduce projective contact manifolds to Fano manifolds with b2 = 1 and projectivized cotangent bundles.
- Low-dimensional cases were settled by Poon and Salamon (real dimension eight) and by Ye and Druel (contact Fano manifolds of complex dimension three and five). Further classifications assume extra structure: Beauville (reductive automorphism group and generically finite contact map), Yasukura (an immersion hypothesis), Buczyński and Wiśniewski (real dimensions twelve and sixteen, and contact Fano manifolds of complex dimension at most nine with reductive automorphism group), Occhetta, Romano, Solá Conde and Wiśniewski (a rank bound on the automorphism group), and Brendle and Semmelmann (nonnegative sectional curvature).
- The manuscript also records that earlier versions of Yasukura's preprint asserted unrestricted classifications that the author later identified as erroneous.
The original results
- Claim. Theorem 1.1 of Contact Fano manifolds and the LeBrun–Salamon conjecture states that every smooth connected complex projective contact Fano manifold of dimension 2n + 1, n ≥ 1, is contact-isomorphic to the adjoint variety of a simple complex Lie algebra, with the contact line bundle corresponding to the hyperplane bundle. Its corollaries state that every contact Fano manifold carries a Kähler–Einstein metric; that a projective contact manifold with b2 = 1 is an adjoint variety and one with b2 ≥ 2 is, as a manifold, the projectivized cotangent bundle of a smooth projective variety; and that every closed connected positive quaternionic-Kähler manifold of real dimension at least eight has parallel curvature and is homothetic to a Wolf space.
- Method, as described: starting from the covering contact lines of Kebekus and Buczyński's corrected theorem on the variety of contact-line directions, a suitable pair of contact lines is smoothed, two vector lifts are extended across the degree-two boundary, and their common contact value descends to a symmetric tensor whose diagonal jets generate the first jets of the contact line bundle, which gives transitivity of contact automorphisms. The Riemannian statement follows through the twistor space and LeBrun's results.
- Source. The manuscript Contact Fano manifolds and the LeBrun–Salamon conjecture at commit
1581d4c. - Formalization. The collection lists no Lean formalization for this result.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 20 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | Z. Hu, K. Liu and X. Wan, Transverse rational curves and the LeBrun–Salamon conjecture, arXiv:2610.10410 | Parallel (independent) |
|
arXiv preprint, version 1 (21 pages). |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_359 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap. The reviewer states that it read every section and checked the main steps by hand, without independently verifying the external citations. Its one minor note is that two key inputs are taken from the literature without proof, the spanning of contact-line directions (Kebekus) and the corrected form of Buczyński's theorem on the variety of directions, and that "given the history of errors in this area" it could not check them against the cited papers. | 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
- J. Daura Serrano, A. Gohr, M.-A. Lawn and T. Schedler, The LeBrun–Salamon conjecture is true in dimension up to 56 (arXiv:2610.06336, version 1, 5 October 2026, the day before the collection's release), state a proof for quaternionic dimension n ≤ 14, by bounding the isometry group of a minimal counterexample and showing that a characteristic number is a sum of positive quantities; its comments mention a partial Lean verification. It predates the collection's release and is not a follow-up.
- The earlier classifications 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.