On the structure of generalized polarized manifolds with relatively small second class

Let X be a smooth complex projective variety of dimension n geq 2 polarized by an ample line bundle H. For n=2, the structure of pairs (X,H) as above is described under the assumption that m-3d leq 4, where m and d stand for the class and the degree of (X,H), respectively. In particular, special attention is payed to the case m-2d leq 1, in which X turns out to be a ruled surface. In higher dimensions, adding to the above setting an ample vector bundle E of rank r leq n-2 on X such that F:=E oplus H^{oplus n-r-2} admits a regular section vanishing on a smooth surface, by relying on the analysis made in the surface case, the structure of the generalized polarized manifold (X,mathcal F) is described for small values of m_2 - 3d, where m_2:=m_2(X,E,H) is the generalized second class defined in a previous paper and d:=c_{n-2}(F) H^2.