Ample vector bundles with zero loci of small $Delta$-genera

Let $X$ be a smooth complex projective variety endowed with an ample vector bundle $mathcal E$ admitting a global section whose zero locus is a smooth subvariety $Z$ of the expected dimension, and let $H$ be an ample line bundle on $X$, whose restriction $H_Z$ to $Z$ is very ample. Triplets $(X, {mathcal E}, H)$ are studied and classified under the assumption that the delta genus of $(Z, H_Z)$ is either small ($leq 3$) or small in comparison with the corank of $mathcal E$ or the degree.