Result 062, Algebraic and complex geometry

Projective contact classification and the LeBrun–Salamon conjecture

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 b2=1b_2=1 are adjoint varieties with their canonical contact structures, while those with b2≥2b_2\ge2 have underlying manifold P(T∗Z)\mathbb P(T^*Z) for a smooth projective variety Z.

Proof

The bigger picture

Why it matters

A contact structure is a field of tangent hyperplanes that cannot fit together into hypersurfaces. The claimed classification says that projective spaces carrying such structures belong to sharply prescribed families, linking complex geometry to rigid Riemannian models.

What changes?

The manuscript claims to classify smooth connected complex projective contact manifolds of complex dimension at least three. If the second Betti number, measuring independent two-dimensional cohomology classes, is one, these are adjoint varieties with canonical contact structures. If it is at least two, the underlying manifold is a projectivized cotangent bundle of a smooth projective variety. It also claims every contact Fano manifold in this setting, defined by an algebraic positivity condition, is an adjoint variety with its given contact structure.

What does that help mathematicians do?

As a consequence, the manuscript claims that every closed connected smooth positive quaternionic-Kähler manifold of real dimension divisible by four and at least eight is a compact symmetric Wolf space up to metric rescaling. Here closed means compact without boundary. This would rule out nonsymmetric examples throughout that class, turning the search for new geometries under these hypotheses into the study of the specified symmetric models.

Are there practical applications?

Its immediate value is foundational. Adjoint varieties are geometric models built from simple complex Lie algebras; projectivized cotangent bundles record cotangent directions over another variety. The classification would direct researchers toward these explicit constructions, while distinguishing cases where the contact structure itself is determined from those where only the underlying manifold is classified.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Contact Fano manifolds and the LeBrun–Salamon conjecture

September 23, 2026 40 pages

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≥84m\geq8 is homothetic to a compact symmetric Wolf space.

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Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.