Ample Vector Bundles with Smallg − q

Let X be a smooth complex projective variety and let Z ⊂ X be a smooth submanifold of dimension ≥ 2, which is the zero locus of a section of an ample vector bundle ℰ of rank dim X − dim Z ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H Z to Z is generated by global sections. The structure of triplets (X,ℰ,H) as above is described under the assumption that the curve genus of the corank-1 vector bundle ℰ ⊕ H ⊕ (dim Z−1) is ≤ h 1( X ) + 2.