Hyperbolicity for log canonical pairs and the cone theorem

Given a log canonical pair $(X, \Delta)$, we show that $K_X+\Delta$ is nef assuming there is no non-constant map from the affine line with values in the open strata of the stratification induced by the non-klt locus of $(X, \Delta)$. This implies a generalization of the Cone Theorem where each $K_X+\Delta$-negative extremal ray is spanned by a rational curve that is the closure of a copy of the affine line contained in one of the open strata of $\mathrm{Nklt}(X, \Delta)$. Moreover, we give a criterion of Nakai type to determine when under the above condition $K_X+\Delta$ is ample and we prove some partial results in the case of arbitrary singularities.