The LeBrun-Salamon conjecture: positive quaternionic-Kahler manifolds and contact Fano manifolds are homogeneous
A Riemannian manifold of dimension , , is quaternionic-Kahler if its holonomy lies in ; it is positive if its scalar curvature is positive. Wolf showed that the compact symmetric examples (the Wolf spaces) correspond to the adjoint varieties of simple complex Lie algebras, and Salamon's twistor construction attaches to every closed positive quaternionic-Kahler manifold a complex contact Fano manifold. LeBrun and Salamon proved finiteness and strong rigidity results and conjectured that the symmetric examples are the only ones. The algebraic form, the contact-Fano homogeneity conjecture, asks the same of every smooth complex projective contact Fano manifold, which should be an adjoint variety . Known cases covered real dimension 8, 12 and 16 and low complex dimensions, or required extra hypotheses (reductive automorphism group, nonnegative sectional curvature). Is every closed connected positive quaternionic-Kahler manifold of dimension at least 8 a Wolf space, and is every contact Fano manifold an adjoint variety?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Differential and algebraic geometry; quaternionic-Kahler and contact Fano
- Posed by
- Claude LeBrun and Simon Salamon
- Year posed
- 1994
- Years open
- 32y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 56 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: every smooth connected complex projective contact Fano manifold of dimension , , is contact-isomorphic to the adjoint variety of a simple complex Lie algebra, with . Corollaries: every such manifold is Kahler-Einstein; every smooth projective contact manifold with is an adjoint variety and with is (via Kebekus-Peternell-Sommese-Wisniewski and Demailly); every closed connected positive quaternionic-Kahler manifold of real dimension satisfies and is homothetic to a Wolf space. It says nothing about negative or zero scalar curvature, about noncompact or incomplete metrics, or about compact Kahler contact manifolds that are not projective beyond the jet criterion it states.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family is a single manuscript (September 23, 2026). It cites earlier claimed proofs by Yasukura and by Kobayashi (2008) and records that Yasukura later identified errors in his.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2-1.4 of the manuscript were read against the conjecture as posed. Theorem 1.1 states that every smooth connected complex projective contact Fano manifold of dimension is contact-isomorphic to an adjoint variety; Corollary 1.4 deduces that every closed connected positive quaternionic-Kahler manifold of real dimension is homothetic to a Wolf space. This is the conjecture in full. The proof (contact-line smoothing, a descended symmetric section on , a jet criterion for contact transitivity) was not refereed, and there is no Lean formalization. Readers should know that at least two earlier claimed proofs of the unrestricted classification (Yasukura's preprint versions, which the author later withdrew in part) did not survive, which the manuscript itself records.