On exterior powers of the tangent bundle on toric varieties

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 ${\mathrm{\Lambda }}^m{\mathcal{T}}_X$ in terms of wall relations. As an application, we classify smooth toric varieties of arbitrary dimension $n\ge 3$ with ${\mathrm{\Lambda }}^2{\mathcal{T}}_X$ nef and those of dimension $n\ge 4$with$\Lambda^3\mathcal T_X$ ample.