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

Nefness of adjoint bundles for ample vector bundles of corank 3