# Projective contact classification and the LeBrun–Salamon conjecture > Contact Fano manifolds are homogeneous, which settles the LeBrun–Salamon conjecture on quaternionic-Kähler spaces. Plain-English explainer with sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-062/ Last updated: 2026-10-14 **Contact Fano manifolds are homogeneous, which settles the LeBrun–Salamon conjecture on quaternionic-Kähler spaces.** **Trust:** Claimed, not checked · **Papers:** 1 · **First paper:** September 23, 2026 · **Field:** Algebraic and complex geometry · **Entry:** 062 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. A contact structure is a field of maximally twisted hyperplanes on an odd-dimensional manifold. The paper claims every smooth projective contact Fano manifold is on a known list built from simple Lie algebras. A corollary is the LeBrun–Salamon conjecture. Every closed positive quaternionic-Kähler manifold of real dimension at least eight would be a symmetric Wolf space, up to scale. The paper also claims a classification of all projective contact manifolds. ## At a glance - **First step:** 1994 (C. LeBrun and S. Salamon, strong rigidity of positive quaternion-Kähler manifolds) - **Significance:** Major. If correct, it settles a decades-old classification problem linking complex contact geometry and Riemannian holonomy. - **Trust:** Claimed, not checked - **Field:** Algebraic and complex geometry (1 paper) ## The explanation Take ordinary three-dimensional space with coordinates x, y and z. Consider the one-form α = dz − y dx. At each point it picks out a plane of directions, the ones on which α is zero. Its derivative is dα = dx ∧ dy. Then α ∧ dα = dz ∧ dx ∧ dy, which is never zero. That is exactly the condition for a contact structure[1]. The planes twist so much that no surface is tangent to all of them, even locally. On a complex manifold of odd dimension 2n + 1, a contact structure is such a field of hyperplanes. A contact Fano manifold is a compact projective one whose anticanonical bundle is ample, a strong positivity condition. There is a family of standard examples. Each simple complex Lie algebra, an algebra of infinitesimal symmetries, has a closed adjoint orbit. Projectivized, it gives the adjoint variety, with a canonical contact structure[2]. For example, three-dimensional projective space arises from the symplectic Lie algebra of 4 × 4 matrices[3]. The contact-Fano homogeneity conjecture says these are the only examples. This paper claims to prove it[4]. Every smooth connected complex projective contact Fano manifold of dimension at least three is one of these adjoint varieties[5]. The match also carries the given contact structure to the canonical one[5]. The second payoff is in Riemannian geometry. A quaternionic-Kähler manifold has real dimension 4m and a metric whose holonomy lies in the group Sp(m)Sp(1)[6]. Holonomy records how directions turn when carried around loops. Positive means positive scalar curvature, a kind of average curvature. The compact symmetric examples are called Wolf spaces[6]. The LeBrun–Salamon conjecture concerns the case m ≥ 2. It says every closed positive quaternionic-Kähler manifold is a Wolf space up to scale. The paper claims to prove this as a corollary[7,8]. For m = 2 the real dimension is 8. The link is the twistor space. Each such manifold of real dimension 4m has a twistor space, a two-sphere bundle over it. Its real dimension is 4m + 2, so its complex dimension is 2m + 1. Salamon showed this twistor space is a contact Fano manifold[9]. So the algebraic theorem applies to it[8]. From the twistor space the paper then recovers the original metric[8]. What was known before? Poon and Salamon settled real dimension eight[10]. Buczyński and Wiśniewski settled real dimensions twelve and sixteen[11]. Several results needed extra hypotheses, such as a reductive symmetry group[12]. Brendle and Semmelmann proved symmetry under nonnegative sectional curvature[13]. Earlier preprints by Kobayashi and Yasukura claimed broad classifications. Later comments by Yasukura identified errors in two of his versions[14]. The obstacle is to find enough symmetries. The paper must build global vector fields that preserve the contact structure and span every tangent space[15]. It does this with contact lines, the simplest rational curves in the manifold. Two such lines are joined and smoothed into a conic[16]. The paper builds a symmetric section on pairs of points whose behaviour near the diagonal forces the fields to span[17]. Then the symmetry group acts transitively, and the manifold is homogeneous. The paper also claims a classification of all projective contact manifolds of dimension at least three. With second Betti number 1, they are adjoint varieties[18]. With second Betti number at least 2, the manifold is the projectivized cotangent bundle of a smooth projective variety[18]. How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published. **What it doesn't show.** This does not cover quaternionic-Kähler manifolds with negative or zero scalar curvature, or real dimension four. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major only if the proof holds. Earlier claimed proofs of this classification were later found to contain errors. No Lean formalization is listed and no specialist review is published. ## Words used - **Contact structure:** A field of hyperplanes that twists so much that no hypersurface is tangent to all of them. - **Fano:** Having an ample anticanonical bundle, a strong form of positive curvature. - **Adjoint variety:** The projectivized smallest nonzero nilpotent orbit of a simple Lie algebra, with its canonical contact structure. - **Quaternionic-Kähler manifold:** A Riemannian manifold of real dimension 4m whose holonomy lies in Sp(m)Sp(1). - **Holonomy:** The way directions turn when carried around closed loops. - **Wolf space:** A compact symmetric positive quaternionic-Kähler space. - **Twistor space:** A two-sphere bundle over a quaternionic-Kähler manifold that carries a complex structure. - **Homothetic:** Equal after rescaling the metric by a constant. ## The original paper - [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf) - September 23, 2026 ## Quoted lines from the paper (18) [1] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > The condition is equivalent to $\alpha\wedge(d\alpha)^n$ being nowhere zero. [2] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > This is the unique closed adjoint orbit in $\PP(\mathfrak g)$. [3] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > In the first, the adjoint embedding is the quadratic Veronese embedding, so its hyperplane bundle is $\OO(2)$. [4] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > Theorem … resolves the smooth contact-Fano homogeneity conjecture positively. [5] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > 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. [6] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > The compact symmetric positive quaternionic-K\"ahler spaces are the \emph{Wolf spaces}. [7] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > Moreover, $(M,g)$ is homothetic to a compact symmetric positive quaternionic-K\"ahler Wolf space. [8] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > The link is the positive twistor construction: the twistor space is contact Fano, so the general algebraic theorem applies to it. [9] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > The twistor construction of Salamon, independently due to B\'erard-Bergery, associates a contact Fano manifold to a closed connected positive quaternionic-K\"ahler manifold. [10] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > Earlier low-dimensional classifications were obtained by Poon and Salamon for compact positive quaternionic-K\"ahler manifolds in real dimension eight, and by Ye and Druel for contact Fano manifolds in complex dimensions three and five, respectively. [11] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > Buczy\'nski and Wi\'sniewski classify the closed positive quaternionic-K\"ahler manifolds in real dimensions twelve and sixteen, and the contact Fano manifolds in complex dimensions three through nine with reductive automorphism group. [12] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > Several classification results impose additional geometric or group-theoretic hypotheses. [13] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > On the Riemannian side, Brendle and Semmelmann establish symmetry for compact positive quaternionic-K\"ahler manifolds of real dimension at least eight under nonnegative sectional curvature. [14] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > The author's later comments identify an error in Theorem 3.2(ii) of the former version, and identify Theorem A and Corollary B of the latter version as false. [15] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > The objective is to produce enough global vector fields preserving $F$ to span every tangent space. [16] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > Their union can be smoothed while its two outer values move. [17] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > This local behavior will force the global contact fields to span. [18] From [Contact Fano manifolds and the LeBrun–Salamon conjecture](https://github.com/openai/math/blob/main/preprints/Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026/paper.pdf): > Thus these are precisely the underlying manifolds of smooth connected complex projective contact manifolds in these dimensions. ## Other sources - [openai/math release notes (2026): model provenance](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” - [C. LeBrun and S. Salamon, “Strong rigidity of positive quaternion-Kähler manifolds”, Inventiones Mathematicae 118 (1994), 109–132](https://doi.org/10.1007/BF01231528) - [S. Salamon, “Quaternionic Kähler manifolds”, Inventiones Mathematicae 67 (1982), 143–171](https://doi.org/10.1007/BF01393378) - [C. LeBrun, “Fano manifolds, contact structures, and quaternionic geometry”, International Journal of Mathematics 6 (1995), 419–437](https://doi.org/10.1142/S0129167X95000146) - [Y. S. Poon and S. M. Salamon, “Quaternionic Kähler 8-manifolds with positive scalar curvature”, Journal of Differential Geometry 33 (1991), 363–378](https://doi.org/10.4310/jdg/1214446322) - [J. Buczyński and J. A. Wiśniewski, “Algebraic torus actions on contact manifolds”, Journal of Differential Geometry 121 (2022), 227–289](https://doi.org/10.4310/jdg/1659987892) - [S. Brendle and U. 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