Projective contact classification and the LeBrun–Salamon conjecture

1 · The essentials

This family concerns whether spaces with certain geometric features must belong to particular standard families. The main manuscript claims a positive answer to two linked classification questions. The first concerns complex geometric spaces with a specified arrangement of allowed directions at each point, together with additional geometric restrictions. It says that every such space satisfying its assumptions matches a standard model, including those specified directions. The second concerns a different class of curved spaces, in real dimension at least eight, and says that each matches a compact symmetric model after a uniform change of scale. The repository summary also reports a broader classification of the complex spaces into two cases. These are manuscript claims: the main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine sorting objects whose shapes and markings must obey a strict set of rules. The first claim says that both the object and its markings match a standard template; the second allows a uniform resizing before making the match. This image does not capture the actual geometric restrictions.

Why it matters

The manuscript claims to settle both the contact-Fano homogeneity conjecture and the Riemannian LeBrun–Salamon conjecture positively. Its first classification preserves the given contact distribution, rather than identifying only the underlying space. The abstract presents the Riemannian classification as a consequence of that contact-geometric result.

2 · Going further

The main contact-geometric claim concerns every smooth connected complex projective contact Fano manifold of complex dimension at least three. With its given contact distribution, each is claimed to be contact-isomorphic to the adjoint variety of a simple complex Lie algebra. Thus the classification identifies the space together with its contact structure, not merely its underlying manifold.

The abstract states the Riemannian consequence as follows: every closed connected smooth positive quaternionic-Kähler manifold of real dimension $4 m \geq 8$ is homothetic to a compact symmetric Wolf space. Here homothety allows a constant rescaling of the metric. The manuscript describes these conclusions as positive resolutions of the contact-Fano homogeneity conjecture and the LeBrun–Salamon conjecture.

The repository summary reports a broader classification of smooth connected complex projective contact manifolds of complex dimension at least three. Those with $b_{2} = 1$ are identified as adjoint varieties with their canonical contact structures; those with $b_{2} \geq 2$ have underlying manifold $P (T^{*} Z)$ for a smooth projective variety $Z$ . The wording distinguishes a contact-structure classification in the first case from an underlying-manifold statement in the second.

The supplied abstract gives the logical connection from contact-Fano homogeneity to the Riemannian conclusion, but does not describe the argument establishing either claim. Only the main manuscript is supplied, so no account of companion manuscripts or their role can be given from this source.

Useful notions

Smooth manifolds and dimension
Complex projective geometry
Contact distributions
Fano manifolds
Riemannian metrics and constant rescaling

What was known before / what is new

Before

The source names two conjectures: contact-Fano homogeneity and the Riemannian LeBrun–Salamon conjecture. It gives no account of earlier partial results or of which cases were previously established.

New

The main manuscript claims the full contact-Fano classification under its stated smoothness, connectedness, projectivity, and dimension assumptions, preserving the given contact distribution. It also claims the positive quaternionic-Kähler classification as a consequence. The repository summary additionally reports the projective contact classification split by $b_{2}$ , although that broader statement is not spelled out in the supplied abstract.

Limits and assumptions

The contact-Fano claim requires a smooth connected complex projective contact Fano manifold of complex dimension at least three. The Riemannian claim requires a closed connected smooth positive quaternionic-Kähler manifold of real dimension $4 m \geq 8$ . In the broader repository classification, the $b_{2} \geq 2$ statement identifies the underlying manifold as $P (T^{*} Z)$ ; it does not explicitly identify the given contact structure. The supplied text contains no argument details or list of remaining open cases, and it makes no classification claim outside the stated hypotheses. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Contact distribution: The arrangement of tangent directions that specifies the contact structure on a contact manifold.
Contact-isomorphic: Equivalent by an isomorphism that also preserves the contact distributions.
Homothetic: Equivalent as metric spaces of the type under discussion after a constant rescaling of the metric.
Adjoint variety: The geometric model associated with a simple complex Lie algebra that appears as the target of the manuscript's contact-Fano classification.
Compact symmetric Wolf space: A member of the family of compact symmetric models appearing in the claimed positive quaternionic-Kähler classification.

Keywords : Projective contact classification · Contact-Fano homogeneity · LeBrun–Salamon conjecture

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Contact Fano manifolds and the LeBrun–Salamon conjecture

23/09/2026

To be verified

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.

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