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Math-472-A2 · revision r1 · on Math-472

OpenAI claim summary for Math-472

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 062, 2026-09-23): Contact-Fano homogeneity implies the Campana–Peternell conclusion for the subset of nef-tangent Fano manifolds admitting a contact structure; this problem imposes no contact hypothesis.

Manuscript 2 (family 067, 2026-09-25): The homogeneity theorem treats exactly complex dimension six; the catalogue Campana–Peternell assertion ranges over all Fano dimensions.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Campana–Peternell Conjecture for Fano Manifolds
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Campana–Peternell Conjecture for Fano Manifolds

OpenAI's claim

Manuscript 1 (family 062, 2026-09-23): 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. Manuscript 2 (family 067, 2026-09-25): Among smooth complex Fano varieties in dimension six, those with nef tangent are claimed to be rational homogeneous, with a universal-cover consequence when dimension minus maximal étale irregularity is at most six.

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.
Mathematical claim 2.
OpenAI claims rational homogeneity for each smooth connected complex projective Fano manifold of dimension six whose tangent bundle is nef (a quotient of a complex semisimple group by a parabolic subgroup). Its Kähler uniformization corollary considers a compact connected Kähler $X$ with analytically nef tangent bundle and \[ 0\le d=\dim X-\widetilde q(X)\le6,\qquad \widetilde q(X)=\max_{X'\to X\text{ finite étale}}h^1(X',\mathcal O_{X'}). \] It claims a connected finite étale cover whose Albanese morphism is a holomorphically locally trivial bundle with rational homogeneous fiber $F$ of dimension $d$, and \[ \widetilde X\simeq F\times\mathbb C^{\widetilde q(X)}. \] Thus the universal cover is quasi-projective and holomorphically convex, with Remmert reduction to the affine factor. In particular the corollary applies in complex dimension at most six.

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.

Manuscript 2 (family 067, 2026-09-25): Known contractions reduce the new case to Picard number one and pseudoindices four or five. The universal minimal-curve family has two smooth fibrations whose projection identities give linear intersection constraints; characteristic-number and integrality comparisons rule out the remaining models and identify the homogeneous possibilities.

Formal verification

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

Family 067 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