LeBrun-Salamon Conjecture: positive quaternion-Kahler manifolds are symmetric spaces

LeBrun-Salamon Conjecture: positive quaternion-Kahler manifolds are symmetric spaces

OpenAI's claim

Every complex contact Fano manifold is claimed to be an adjoint variety with its given contact distribution; the quaternionic-Kähler consequence identifies positive manifolds of real dimension $4m$, $m\ge2$, with symmetric Wolf models up to scale. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

For every smooth connected projective complex contact Fano manifold $(X,F)$ of dimension $2n+1$, $n\ge1$, with contact line $L=T_X/F$, OpenAI claims a simple complex Lie algebra $\mathfrak g$ and an algebraic contact isomorphism \[ X\simeq\mathbb P(\mathcal O_{\min}(\mathfrak g)),\qquad L\simeq\mathcal O_{\mathbb P(\mathfrak g)}(1)|_X. \] The target is the projectivized least-dimensional nonzero nilpotent adjoint orbit, with its canonical contact distribution. The Riemannian corollary says that every closed connected $4m$-manifold, $m\ge2$, with holonomy contained in $\operatorname{Sp}(m)\operatorname{Sp}(1)$ and positive scalar curvature has $\nabla\operatorname{Rm}=0$ and agrees up to constant metric scaling with a Wolf space, which is compact, symmetric and positive quaternionic-Kähler.

Outline of the argument

Manuscript 1 (family 062, 2026-09-23): Two contact lines with nonzero Levi pairing are smoothed into very free conics while tracking both endpoint velocities. This produces a symmetric line-valued section on the square with the prescribed diagonal jet, forcing global contact fields to span; homogeneity follows, and the twistor correspondence supplies the Riemannian conclusion.

Formal verification

Family 062 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

• OpenAI, Contact Fano manifolds and the LeBrun–Salamon conjecture, 2026-09-23; citation key OAI:Contact-Fano-manifolds-and-the-LeBrun-Salamon-conjecture-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: thm:quaternionic.
• OpenAI catalogue entry.

LeBrun-Salamon Conjecture: positive quaternion-Kahler manifolds are symmetric spaces

Definitions and mathematical statement

In real dimension $4n$, $n\ge2$, a quaternion-Kähler manifold is a Riemannian manifold whose holonomy lies in $\operatorname{Sp}(n)\operatorname{Sp}(1)$. Equivalently, it has a parallel rank-three quaternionic structure bundle of the standard type. Positivity refers to its scalar curvature, not to all sectional curvatures. The conjecture asks whether every compact positive quaternion-Kähler manifold is a Riemannian symmetric space, as in the cited classification formulation.

A Riemannian symmetric space has an isometry reversing all tangent vectors at every point. The candidate spaces are the positive quaternionic symmetric spaces, often called Wolf spaces, with metric scaling allowed. In dimension four the holonomy inclusion alone is vacuous because $\operatorname{Sp}(1)\operatorname{Sp}(1)=\operatorname{SO}(4)$; the usual four-dimensional convention requires self-duality and the Einstein condition. Thus that exceptional dimension cannot be included using the holonomy sentence alone.

Formulation and concept references: [S1].

Short English statement

Are compact quaternion-Kähler manifolds of positive scalar curvature necessarily among the symmetric examples?

Research attempt

Scope, attribution, and source correction.

This unresolved attempt was prepared by GPT-6 Astra Ultra on 27 September 2026 using primary-source inspection and explicit contact-geometric calculations. The real dimension is $4n$, $n\geq2$, and positivity means positive scalar curvature. The inherited source entry [S1] is preserved verbatim, but its URL resolves to the paper by Udhav Fowdar and Henrique Sá Earp, rather than the authors named in that entry. The linked paper concerns a harmonic flow of quaternionic structures; it does not classify every positive quaternion-Kähler metric. For the geometric reduction we use LeBrun's primary paper [E1]. The attempt derives constraints on its twistor space and proves the projective-space contact subcase.

The established twistor input.

A positive quaternion-Kähler manifold has a twistor space $\pi:Z\to M$ with fiber $\mathbb P^1$, complex dimension $2n+1$, a holomorphic contact structure and a positive Kähler--Einstein metric. These are established geometric results used here. Conversely the contact Kähler--Einstein construction in [E1] recovers a positive quaternion-Kähler manifold. Thus a classification of the relevant contact Kähler--Einstein spaces, with their geometric reconstruction, would give a classification route for the original metric problem.

This is a reduction, not a claim that every Fano manifold is homogeneous or that every complex contact manifold automatically carries such a metric. The Kähler--Einstein and contact hypotheses must travel together. The original target also concerns the Riemannian metric, so identifying a smooth manifold alone would be insufficient.

A determinant identity from the contact form.

Let $D\subset TZ$ be the rank-$2n$ contact distribution and $L=TZ/D$ its quotient line bundle. Locally the quotient map is an $L$-valued one-form $\theta$. Although $d\theta$ depends on a trivialization of $L$, its restriction to $D$ does not: under $\theta\mapsto f\theta$ the term $df\wedge\theta$ vanishes on $D$. Contact nondegeneracy says that $(d\theta)|_D$ is a nondegenerate alternating form with values in $L$.

Taking its $n$-th exterior power gives a nowhere-zero section of $(\det D)^*\otimes L^n$, hence $\det D\cong L^n$. The exact sequence $0\to D\to TZ\to L\to0$ now gives \begin{equation}\label{wsref-0-61-34-34-32-3a-63-61-6e-6f-6e-69-63-61-6c} K_Z^{-1}\cong L^{n+1}. \end{equation} This is an isomorphism of line bundles, not merely an equality of real Chern classes with possible torsion suppressed. Equivalently $\theta\wedge(d\theta)^n$ trivializes $K_Z\otimes L^{n+1}$.

For a twistor space, $K_Z^{-1}$ is ample, so $L$ is ample too: a positive tensor power of a line bundle is ample precisely when the line bundle is ample. For every curve $C$, $(-K_Z)\cdot C=(n+1)L\cdot C$. These divisibility and positivity restrictions are necessary, but they do not classify $Z$. Many integer intersection data can satisfy a divisibility condition without determining a variety.

A complete algebraic subcase: projective twistor space.

Assume additionally that $Z\cong\mathbb P^{2n+1}$. Its Picard group is generated by $\mathcal O(1)$ and $K_Z^{-1}=\mathcal O(2n+2)$. Equation (a442:canonical) forces $L=\mathcal O(2)$. The Euler sequence, twisted by $\mathcal O(1)$, identifies \[ H^0(\mathbb P(V),\Omega^1(2)) =\ker\left(V^*\otimes V^*\longrightarrow \operatorname{Sym}^2V^*\right) =\Lambda^2V^*, \qquad\dim V=2n+2. \] Thus a contact quotient form is represented by an alternating bilinear form $A$ on $V$.

If $A$ has a nonzero radical vector $z$, the quotient form vanishes at $[z]$ and cannot define a contact distribution there. If $A$ is nondegenerate, its induced form on $z^{\perp_A}/\mathbb Cz$ is nondegenerate; this is precisely the contact nondegeneracy on the rank-$2n$ distribution. Therefore the contact forms are exactly the nondegenerate alternating forms.

All such forms on a fixed even-dimensional complex vector space are equivalent under $\operatorname{GL}(V)$, by the elementary symplectic-basis construction. Consequently all contact structures of this form on projective space are equivalent under projective automorphisms. Combining this explicit classification with the established twistor reconstruction and uniqueness up to metric scale gives the standard quaternionic projective space $\mathbb H P^n$. The additional assumption $Z\cong\mathbb P^{2n+1}$ is the entire restriction; it has not been deduced for general $M$, and the other Wolf spaces would not satisfy it.

A topological check from the twistor fibration.

The contact line restricts to $\mathcal O(2)$ on each twistor line. Thus $h=\tfrac12c_1(L)\in H^2(Z,\mathbb Q)$ restricts to the generator of $H^2(\mathbb P^1,\mathbb Q)$. The classes $1,h$ restrict to a basis of the fiber cohomology. Leray--Hirsch therefore gives, as graded rational vector spaces, \[ H^k(Z,\mathbb Q)\cong H^k(M,\mathbb Q)\oplus H^{k-2}(M,\mathbb Q). \] In particular $b_2(Z)=b_2(M)+1$. This argument uses an actual global class restricting to the fiber generator; it does not simply assume that every sphere bundle is a cohomological product.

For the projective case, $b_2(Z)=1$, so $b_2(M)=0$, consistent with $\mathbb H P^n$. The converse inference would be invalid: $b_2(M)=0$ gives $b_2(Z)=1$, not an identification of $Z$ with projective space. Picard number one is much weaker than a complete holomorphic classification. Nor does the vector-space decomposition assert a product ring structure or a topologically trivial fibration.

Why immediate flow and curvature arguments stop.

Let $\Omega$ be the parallel fundamental quaternionic four-form of an actual quaternion-Kähler metric. A torsion energy of the form $\tfrac12\int_M|\nabla\Omega|^2$ is zero. Its negative gradient therefore vanishes at this structure: a differentiable nonnegative functional has zero derivative at a zero minimum. Running such a flow starting from the unknown torsion-free structure does not move it toward a symmetric example. One still has to classify the zero-energy structures. The linked harmonic flow paper also distinguishes harmonic structures from torsion-free ones; merely reaching a critical point would not resolve that issue.

Positive quaternion-Kähler metrics are Einstein, so one may normalize $\operatorname{Ric}=(4n-1)g$. Myers' theorem then bounds the diameter by $\pi$. This is useful geometric control, but it is not a bound on the full curvature tensor or a proof that $\nabla\operatorname{Rm}=0$. The latter is the local symmetry condition that a classification argument would need to establish or replace. Even among the symmetric candidate spaces, higher rank permits zero-curvature planes; positive scalar curvature must not be strengthened to positive sectional curvature.

Thus an Einstein normalization, a diameter bound and the contact Chern relation are compatible necessary restrictions, but none eliminates all hypothetical nonsymmetric examples. Compactness or finite lists of topological data alone would still require a rigidity argument for the metric.

Verification, first gap, and outcome.

The contact exponent $n+1$ was checked from the rank-$2n$ distribution, and the projective calculation gives $\mathcal O(2)$ rather than $\mathcal O(1)$. The rational fiber generator retains its factor of one half. Metric scale is allowed throughout, and the exceptional four-dimensional holonomy convention is not used to enlarge the target.

The first remaining classification input is to show that every twistor contact Kähler--Einstein space in this scope is one of the homogeneous models, or to construct a valid nonhomogeneous one with its required reconstruction. UNRESOLVED. The determinant identity, projective contact classification and topology check are proved. They do not classify all positive quaternion-Kähler manifolds, and no novelty is claimed.

Sources

• S1 Fadel, Loubeau, and Moreno, Harmonic Flow of Quaternion-Kahler Structures, 2024, Introduction. https://link.springer.com/article/10.1007/s12220-024-01624-7. Formulation source.
• E1 Claude LeBrun, Fano Manifolds, Contact Structures, and Quaternionic Geometry, International Journal of Mathematics 6 $1995$, 419--437. https://www.math.stonybrook.edu/~claude/fcq.pdf. Primary author text for the twistor/contact Kähler--Einstein correspondence; consulted 27 September 2026. The inherited [S1] author mismatch is identified in the attempt rather than silently changing its source entry.