Ample vector bundles of small Delta-genera

A natural notion of ``delta-genus'' $Delta$ for a generalized polarized manifold $(X,{mathcal E})$, strictly related to its associated scroll, is introduced and pairs $(X,{mathcal E})$ with low $Delta$ are classified. The stronger are the properties enjoyed by the vector bundle ${mathcal E}$, the larger are the values of $Delta$ attained by the results.