The Campana-Peternell conjecture on Fano manifolds with nef tangent bundle, in dimension six

Theorem 1.1: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous. The new step is Picard number one and pseudoindex five, where a characteristic-number identity $32-20u+3u^2=0$ forces the VMRT to be the Segre threefold $\mathbb P^1\times\mathbb P^2\subset\mathbb P^5$, so $X\cong\mathrm{Gr}(2,5)$. Corollary 1.2: a compact Kahler manifold with nef tangent bundle and $\dim X-\tilde q(X)\le6$ has universal cover $F\times\mathbb C^{\tilde q(X)}$ with $F$ rational homogeneous. Dimensions seven and higher remain open.

Theorem 1.1: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous. The new step is Picard number one and pseudoindex five, where a characteristic-number identity $32-20u+3u^2=0$ forces the VMRT to be the Segre threefold $\mathbb P^1\times\mathbb P^2\subset\mathbb P^5$, so $X\cong\mathrm{Gr}(2,5)$. Corollary 1.2: a compact Kahler manifold with nef tangent bundle and $\dim X-\tilde q(X)\le6$ has universal cover $F\times\mathbb C^{\tilde q(X)}$ with $F$ rational homogeneous. Dimensions seven and higher remain open.