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Math-2393 · mathematics · Geometry

Homogeneity of Fano Contact Manifolds

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Must every smooth Fano contact manifold be homogeneous? Fano means that the anticanonical bundle is ample; a contact structure is a maximally nonintegrable holomorphic hyperplane distribution. Homogeneous means that a complex algebraic group acts transitively.

Formulation references: [S1], [S2].

Short English statement

Must every smooth Fano contact manifold be homogeneous? Fano means that the anticanonical bundle is ample; a contact structure is a maximally nonintegrable holomorphic hyperplane distribution. Homogeneous means that a complex algebraic group acts transitively.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2393-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2393

The claim identifies every smooth projective contact Fano manifold of dimension at least three, with its contact distribution, with a homogeneous adjoint variety. This implies the recorded homogeneity assertion. In complex dimension one the Fano curve is $\mathbb P^1$, which is homogeneous as well. The additional contact-line identification is compatible with, and stronger than, homogeneity alone.

Homogeneity of Fano Contact Manifolds

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

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