Campana–Peternell conjecture for Fano manifolds with nef tangent bundle

Prove that every Fano manifold with nef tangent bundle is rational homogeneous, equivalently of the form G/P for a semisimple complex Lie group G and a parabolic subgroup P.

Background

The Campana–Peternell conjecture concerns the classification of Fano manifolds whose tangent bundles are nef. Rational homogeneous manifolds provide the expected examples. The conjecture is known in several low-dimensional cases, including dimensions at most five as noted in the paper, but is not known in general. Its resolution would imply the positivity of the Euler characteristic needed in the paper’s discussion of simultaneous positivity for compact Kähler manifolds with nef tangent bundle.