Ample vector bundles with zero loci of sectional genus two

Let X be a smooth complex projective variety and let Z \subset X be a smooth submanifold of dimension \ge 2, which is the zero locus of a section of an ample vector bundle E of rank dim(X)-dim(Z) \ge 2 on X. Let H be an ample line bundle on X, whose restriction H_Z to Z is generated by global sections. Triplets (X,E,H) as above are classified under the assumption that (Z,H_Z_) is a polarized manifold of sectional genus 2. This can be regarded as a step toward the classification of ample vector bundles of corank 1 and curve genus 2.