Manifolds polarized by vector bundles

Let $X$ be a complex projective manifold of dimension $n$ and let $mathcal E$ be an ample vector bundle of rank $r$. Let also $ au = au (X, {mathcal E}) = min{t in {mathbb R} : K_X + t det {mathcal E} { m is nef}}$ be the nef value of the pair $(X, {mathcal E})$. In this paper we classify the pairs $(X, {mathcal E})$ such that $ au (X, {mathcal E}) geq {n-2 over r}$.