On exterior powers of the tangent bundle on toric varieties

Abstract: We study the positivity of exterior powers of the tangent sheaf on toric varieties in order to generalize results by Campana and Peternell about 3-folds with nef second exterior power of the tangent bundle. Using the theory of equivariant vector bundles and the toric MMP, we establish in the smooth case a criterion for the positivity of $\Lambda^m\mathcal T_X$ in terms of wall relations. As an application, we classify smooth toric varieties of arbitrary dimension $n\ge3$ with $\Lambda^2\mathcal T_X$ nef and those of dimension $n\ge 4$ with $\Lambda^3\mathcal T_X$ ample.