The Campana–Peternell conjecture in dimension six

1 · The essentials

The manuscript studies complex geometric spaces and asks whether certain conditions force them to have a highly structured form. Its main result concerns spaces with six complex dimensions. These spaces must satisfy several specific geometric conditions; the claim is not about every six-dimensional space. Under those conditions, the manuscript shows that the space belongs to the class called rational homogeneous spaces. It presents this as resolving the six-dimensional case of the Campana–Peternell conjecture. The abstract also gives a related description of certain spaces after passing to a covering space. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine sorting objects using a checklist and discovering that every object meeting all the requirements belongs to one structured family. Here, the geometric assumptions form the checklist, and the manuscript identifies the family that every qualifying space must belong to.

Why it matters

The result shown in the manuscript settles the stated conjecture in complex dimension six, subject to all its hypotheses. Its consequence also links a condition on a compact Kähler manifold to a precise product description of its ordinary universal cover.

2 · Going further

The precise main statement is that every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous. Each qualifier is part of the hypothesis. The manuscript presents this statement as the resolution of the Campana–Peternell conjecture in complex dimension six.

The abstract also states a consequence for a connected compact Kähler manifold $X$ with nef holomorphic tangent bundle. If $dim_{C} X - q (X) \leq 6$ , then its ordinary universal cover has the form $F \times C^{q (X)}$ , where $F$ is a rational homogeneous manifold. Here $q (X)$ is the maximal irregularity of a connected finite étale cover of $X$ . The bound is therefore on the difference between two quantities, rather than simply on the dimension of $X$ .

The supplied abstract states the main result and this consequence but does not describe the argument. It therefore provides no basis for explaining a proof strategy or intermediate steps. Only one manuscript is supplied, so no relationship to companion manuscripts can be described.

Useful notions

Complex dimension
Smooth complex projective manifolds
Fano manifolds and nef tangent bundles
Compact Kähler manifolds
Universal covers and finite étale covers

What was known before / what is new

Before

The source identifies the problem as the Campana–Peternell conjecture and singles out its six-dimensional case. It gives no account of earlier partial results, previously settled dimensions, or historical development.

New

The manuscript shows that a smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous. It also states the universal-cover description $F \times C^{q (X)}$ for connected compact Kähler manifolds with nef holomorphic tangent bundle satisfying $dim_{C} X - q (X) \leq 6$ .

Limits and assumptions

The main statement requires smoothness, connectedness, complex projectivity, the Fano condition, complex dimension six, and a nef tangent bundle. The covering-space consequence has its own Kähler, connectedness, tangent-bundle, and numerical hypotheses. The source does not state a resolution beyond this scope or identify what remains open. It supplies no details of the argument. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Sixfold: A space of complex dimension six.
$q (X)$: The maximal irregularity of a connected finite étale cover of $X$ .
$F$: The rational homogeneous manifold appearing as a factor in the stated universal-cover description.
$dim_{C} X$: The complex dimension of $X$ .

Keywords : Campana–Peternell conjecture · Nef tangent bundle · Rational homogeneous manifolds

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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The Campana–Peternell conjecture in dimension six

25/09/2026

To be verified

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 $dim_{C} X - q (X) \leq 6$ has ordinary universal cover $F \times C^{q (X)}$ , where F is a rational homogeneous manifold and $q (X)$ is the maximal irregularity of a connected finite étale cover.

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