The Campana–Peternell conjecture in dimension six

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

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What the papers state

Theorem 1.1

In The Campana–Peternell conjecture in dimension six, 25 September 2026

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Every smooth connected complex projective Fano variety of dimension six with nef tangent bundle is rational homogeneous.

Corollary 1.2 (Albanese fibres and uniformization)

In The Campana–Peternell conjecture in dimension six, 25 September 2026

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Let X be a connected compact Kähler manifold whose holomorphic tangent bundle is nef in the analytic sense. Suppose

0\leq d=\dim_{\mathbb C}X-\widetilde q(X)\leq6.

Then there is a connected finite étale cover X'\to X with the following properties.

1.
The Albanese map \alpha:X'\to A=\operatorname{Alb}(X') is a smooth proper surjection with connected fibres, where A is a complex torus of dimension \widetilde q(X). Every fibre is rational homogeneous of dimension d, with a point allowed when d=0.
2.
There is one such fibre F for which \alpha is a holomorphically locally trivial F-bundle. The ordinary universal cover of X is biholomorphic to
\widetilde X\simeq F\times\mathbb C^{\widetilde q(X)}.
It is therefore biholomorphic to a quasi-projective variety and is holomorphically convex. Its Remmert reduction is the projection
F\times\mathbb C^{\widetilde q(X)} \longrightarrow\mathbb C^{\widetilde q(X)}.

In particular, these conclusions hold whenever \dim_{\mathbb C}X\leq6.

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The Campana–Peternell conjecture in dimension six

25 September 2026, current edition.

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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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Lemma 5.1 of The Campana–Peternell conjecture in dimension six, 25 September 2026
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cert-machine
Method
Exact computation
Their verdict
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: Lemma lem:finite-ratio (from the matrices A, C_1, N defined in Section 4, N(xi_i eta_j) = 0 with xi_0 = eta_0 = 1 forces 32 - 20 eta_1 + 3 eta_1^2 = 0); dim_Q K_3 <= 6 by a rank mod 10007 (a certified lower bound on the rational rank), everything else exact over Q; not the geometry producing these equations, nor the classification”
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30 references in The Campana–Peternell conjecture in dimension six

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