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

Let X be a smooth complex projective variety endowed with an ample vector bundle 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,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 E or the degree.