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Contact Fano manifolds and the LeBrun–Salamon conjecture
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Abstract
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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Follow-up works
- Zhengyu Hu, Kefeng Liu, Xueyuan Wan, Transverse rational curves and the LeBrun–Salamon conjecture (arXiv:2610.10410, 7 Oct 2026). Claims a proof of the Riemannian LeBrun–Salamon conjecture in real dimension 4n, n ≥ 2 (the statement of this paper's Corollary on quaternionic-Kähler manifolds), using balanced transverse rational curves of contact degree two, a two-point tensor, additive group actions, VMRT arguments (Hwang–Mok) and Beauville's theorem; its Remark 1.2 contrasts this manuscript's smoothing of contact lines into very free conics with its own construction, states that the authors' "main result and core argument were obtained well before the appearance of the OpenAI manuscript", and the paper declares use of ChatGPT as an auxiliary tool.
Related prior literature mentioned by others
- Jarosław Buczyński and Jarosław A. Wiśniewski, with an appendix by Andrzej Weber, Algebraic torus actions on contact manifolds (arXiv:1802.05002, 14 February 2018; Journal of Differential Geometry 121, 2022). The AI Proved What? exposition identifies this as the earlier classification through real dimension 16 and a source of contact-Fano classifications under automorphism-group hypotheses.
- Simon Brendle and Uwe Semmelmann, Quaternionic-Kähler manifolds with nonnegative sectional curvature (arXiv:2506.22181, 27 June 2025). The AI Proved What? exposition highlights this earlier symmetry theorem with the additional nonnegative-sectional-curvature assumption, which the OpenAI paper's Riemannian statement does not impose.
Possibly relevant literature (found by the agent, not checked)
- Jordi Daura Serrano, Aron Gohr, Marie-Amélie Lawn, Travis Schedler, The LeBrun–Salamon conjecture is true in dimension up to 56 (arXiv:2610.06336, 5 Oct 2026; supplementary code). Concurrent, independent partial result proving the quaternionic-Kähler statement of this paper's Corollary for quaternionic dimension n ≤ 14 via symmetry-rank bounds and characteristic-number positivity; its supplementary Lean code proves conditional metric symmetry through quaternionic dimension 14 assuming explicit literature inputs, and it does not cite this manuscript.
- Jarosław Buczyński and Giovanni Moreno, Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systems (Banach Center Publ. 117 (2019), 145–176). Survey of the partial classification results for projective contact manifolds (contact Fano manifolds and projectivised cotangent bundles) and of contact VMRT structures, i.e. the prior literature summarized in this paper's history section and the setting of its Corollary on projective contact manifolds.
Digests and expositions
- AI Proved What?, Only Wolf spaces? The LeBrun–Salamon conjecture and contact Fano manifolds (episode 062, October 2026). This video and written transcript explain the manuscript's passage from contact lines to conics, a two-point section, first jets and contact homogeneity, together with its twistor-space consequences. [AI-generated per source]
News and blogs
- Post-Cutoff (AI agents, human editor Adam Bicz), Two claimed proofs of the LeBrun–Salamon conjecture (postcutoff.com, 8 Oct 2026; updated 13 Oct 2026). Places this manuscript alongside the Hu–Liu–Wan preprint arXiv:2610.10410, including the latter's independence statement and disclosure of ChatGPT assistance, and notes that neither has been refereed. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 062
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 062: 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 $b_2=1$ 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.
This is the only paper in this family.