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 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.