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The Campana–Peternell conjecture in dimension six

OpenAI · September 25, 2026

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Abstract

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 $\dim_{\mathbb C}X-\widetilde q(X)\leq6$ has ordinary universal cover $F\times\mathbb C^{\widetilde q(X)}$, where F is a rational homogeneous manifold and $\widetilde q(X)$ is the maximal irregularity of a connected finite étale cover.

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Possibly relevant literature (found by the agent, not checked)

Roberto Muñoz, Gianluca Occhetta, Luis E. Solá Conde, Kiwamu Watanabe and Jarosław A. Wiśniewski, A survey on the Campana-Peternell Conjecture (arXiv:1407.6483, 2014; Rend. Istit. Mat. Univ. Trieste 47 (2015), 127–185). Surveys the conjectured classification of Fano manifolds with nef tangent bundle and its partial results, relevant to the introduction's account of previously known cases, which cites only the original papers.
S. Kebekus, Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron (arXiv:math/0107069, July 2001). Gives a short proof that projective space is characterized by its rational curves; it bears on the pseudoindex-at-least-six step of the pseudoindex reduction proposition, where the paper cites Dedieu–Höring.
Gianluca Occhetta, Luis E. Solá Conde and Eleonora A. Romano, Manifolds with two projective bundle structures (arXiv:2001.06215, 2020; Proc. Amer. Math. Soc. 150, 2022). Classifies Picard-number-two varieties with two mutually uniform projective bundle structures and obtains a Campana–Peternell result under additional group-action hypotheses, a point of comparison for the manuscript's use of two fibrations to identify homogeneous geometry.

Elsewhere

VibeMathed: Problem record
Emergent Mind: family 067

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Cite

@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}
}