Nefness of adjoint bundles for ample vector bundles of corank 3

Let $\mathcal{E}$ be an ample vector bundle of rank $r\geq 2$ on a smooth complex projective variety $X$ of dimension $n$. The aim of this paper is to describe the structure of pairs $(X,\mathcal{E})$ as above whose adjoint bundles $K_X+\det\mathcal{E}$ are not nef for $r=n-3$. Furthermore, we give some immediate consequences of this result in adjunction theory.