Projective contact classification and the LeBrun–Salamon conjecture

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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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What the papers state

Theorem 1.1

In Contact Fano manifolds and the LeBrun–Salamon conjecture, 23 September 2026

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Let (X,F) be a smooth connected complex projective contact Fano manifold of dimension 2n+1, with n\geq1, and let L=T_X/F. There exist a finite-dimensional simple complex Lie algebra \mathfrak g and an algebraic isomorphism

\varphi:X\xrightarrow{\ \sim\ }Y_{\mathfrak g}

whose differential carries the given distribution F to the canonical contact distribution on Y_{\mathfrak g}. Moreover,

L\simeq\varphi^*\bigl(\OO_{\PP(\mathfrak g)}(1)|_{Y_{\mathfrak g}}\bigr).

Corollary 1.3 (Projective contact manifolds)

In Contact Fano manifolds and the LeBrun–Salamon conjecture, 23 September 2026

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

Let (X,F) be a smooth connected complex projective contact manifold of dimension 2n+1, where n\geq1. Its second Betti number determines the following alternatives.

1.
If b_2(X)=1, then (X,F) is contact-isomorphic to the adjoint variety Y_{\mathfrak g}, with its canonical contact distribution, for a simple complex Lie algebra \mathfrak g.
2.
If b_2(X)\geq2, then there is a smooth connected complex projective manifold Z of dimension n+1 and a biholomorphism
X\simeq\PP(T^*Z).
Here \PP(T^*Z) parametrizes lines in cotangent spaces, equivalently hyperplanes in tangent spaces. This assertion identifies the underlying manifolds, without imposing compatibility with the given distribution F.

Conversely, every adjoint variety of complex dimension 2n+1 and every projectivized cotangent bundle in the second alternative admits a contact structure. Thus these are precisely the underlying manifolds of smooth connected complex projective contact manifolds in these dimensions.

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Two more possible matches

Corollary 1.4 in Contact Fano manifolds and the LeBrun–Salamon conjecture, 23 September 2026.
Corollary 1.2 (Kähler–Einstein existence) in Contact Fano manifolds and the LeBrun–Salamon conjecture, 23 September 2026.

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Papers

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Contact Fano manifolds and the LeBrun–Salamon conjecture

23 September 2026, current edition.

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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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37 references in Contact Fano manifolds and the LeBrun–Salamon conjecture

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What people have said

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An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source

Zhengyu Hu, Kefeng Liu, Xueyuan Wan, Transverse rational curves and the LeBrun--Salamon conjecture, 7 October 2026. Source
Hu, Liu and Wan state their LeBrun-Salamon proof was obtained independently and by a different incidence-based route than OpenAI’s smoothing and diagonal-jet argument.

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