  On a smooth complex projective variety $X$ of dimension $n$, consider an
ample vector bundle $\mathcal{E}$ of rank $r \leq n-2$ and an ample line bundle
$H$. A numerical character $m_2=m_2(X,\mathcal{E},H)$ of the triplet
$(X,\mathcal{E},H)$ is defined, extending the well-known second class of a
polarized manifold $(X,H)$, when either $n=2$ or $H$ is very ample. Under some
additional assumptions on $\mathcal{F}: = \mathcal{E} \oplus H^{\oplus
(n-r-2)}$, triplets $(X,\mathcal{E},H)$ as above whose $m_2$ is small with
respect to the invariants $d:=c_{n-2}(\mathcal{F})H^2$ and
$g:=1+\frac{1}{2}\big(K_X + c_1(\mathcal{F})+H\big) \cdot c_{n-2}(\mathcal{F})
\cdot H$ are studied and classified.
