∑math20

Algebraic and complex geometry · Result 062

Projective contact classification and the LeBrun–Salamon conjecture

1 paper

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 \(b_2\ge2\) have underlying manifold \(\mathbb P(T^*Z)\) for a smooth projective variety Z.

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First page of Contact Fano manifolds and the LeBrun–Salamon conjecture

Contact Fano manifolds and the LeBrun–Salamon conjecture

OpenAI · September 23, 2026

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 \(4m\geq8\) is homothetic to a compact symmetric Wolf space.

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@misc{OAI:Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{Contact Fano manifolds and the LeBrun--Salamon conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf}{OAI:Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026}},
  year = {2026}
}

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