Result 067, Algebraic and complex geometry

The Campana–Peternell conjecture in dimension six

Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.

Proof

The bigger picture

Why it matters

A positivity condition on tangent directions can force a complex geometric space to be highly symmetric. The manuscript reports this rigidity in complex dimension six, linking a condition measured along curves to the space's global structure.

What changes?

The manuscript claims that every smooth, connected, complex projective Fano manifold of complex dimension six with nef tangent bundle is rational homogeneous. Fano means its anticanonical line bundle is positive. Nef means that, after pulling the tangent bundle back to any smooth projective curve, every line-bundle quotient has nonnegative degree. Rational homogeneous spaces have a transitive action by a semisimple algebraic group. This establishes the stated classification in dimension six, not in arbitrary dimensions.

What does that help mathematicians do?

The reported consequence also describes connected compact Kähler manifolds with nef holomorphic tangent bundle. Let q be the largest number of independent holomorphic one-forms on any connected finite unbranched cover. If the complex dimension minus q is at most six, the ordinary universal cover is a product of a rational homogeneous manifold and complex Euclidean space of dimension q. This identifies the simply connected geometry explicitly, separating a homogeneous factor from a complex vector-space factor.

Are there practical applications?

Its immediate value is foundational: the claimed classification rules out nonhomogeneous Fano sixfolds satisfying these positivity assumptions. Researchers studying such spaces can therefore focus on rational homogeneous models rather than search for additional types. The covering-space consequence extends that structural control to the specified class of compact Kähler manifolds, without asserting a practical computational method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Campana–Peternell conjecture in dimension six

September 25, 2026 27 pages

We prove that every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous, resolving the Campana–Peternell conjecture in complex dimension six. As a consequence, a connected compact Kähler manifold X with nef holomorphic tangent bundle and dim⁡CX−q~(X)≤6\dim_{\mathbb C}X-\widetilde q(X)\leq6 has ordinary universal cover F×Cq~(X)F\times\mathbb C^{\widetilde q(X)}, where F is a rational homogeneous manifold and q~(X)\widetilde q(X) is the maximal irregularity of a connected finite étale cover.

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Cite (BibTeX) 
@misc{OAI:The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026,
  author = {{OpenAI}},
  title = {{The Campana--Peternell conjecture in dimension six}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf}{OAI:The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.