The Campana–Peternell conjecture in dimension six

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