OpenAI · 062-01 · 2026-09-23 · Algebraic and complex geometry

Contact Fano manifolds and the LeBrun–Salamon conjecture

preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026

Family 062. The LeBrun–Salamon conjecture and projective contact classification.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves the LeBrun–Salamon conjecture: every closed connected positive quaternionic-Kähler manifold of real dimension at least eight is homothetic to a compact symmetric Wolf space. It also proves contact-Fano homogeneity and classifies smooth connected complex projective contact manifolds of complex dimension at least three: those with $b_{2} = 1$ are adjoint varieties with their canonical contact structures, while those with $b_{2} \geq 2$ have underlying manifold $P (T^{*} Z)$ for a smooth projective variety $Z$ .

We resolve the contact-Fano homogeneity conjecture and the Riemannian LeBrun–Salamon conjecture positively. Every smooth connected complex projective contact Fano manifold of complex dimension at least three, with its given contact distribution, is contact-isomorphic to the adjoint variety of a simple complex Lie algebra. Consequently, every closed connected smooth positive quaternionic-Kähler manifold of real dimension $4 m \geq 8$ is homothetic to a compact symmetric Wolf space.

Preprint source and Lean scope

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