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Let $X$ be a complex smooth projective variety such that the exterior powerof the tangent bundle $\bigwedge^{r} T_X$ is nef for some $1\leq r<\dim X$. Weprove that, up to an \'etale cover, $X$ is a Fano fiber space over an Abelianvariety. This gives generalizations of the structure theorem of varieties withnef tangent bundle by Demailly, Peternell and Schneider and that of varietieswith nef $\bigwedge^{2} T_X$ by the author. Our result also gives an answer toa question raised by Li, Ou and Yang for varieties with strictly nef$\bigwedge^{r} T_X$ when $r < \dim X$.

Positivity of the exterior power of the tangent bundles