Generalized polarized manifolds with low second class

Abstract

On a smooth complex projective variety $X$ of dimension $n$, consider an ample vector bundle $\mathscr E$ of rank $r \leq n - 2$ and an ample line bundle $H$. A numerical character $m_2 = m_2(X,\mathscr E, H)$ of the triplet $(X,\mathscr E, H)$ 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 $\mathscr F:= \mathscr E \oplus H^{\oplus(n-r-2)}$, triplets $(X,\mathscr E, H)$ as above whose $m_2$ is small with respect to the invariants $d := c_{n-2}(\mathscr F)H^2$ and $g := 1 + \frac{1}{2}(K_X + c_1(\mathscr F)+H)\cdot c_{n-2}(\mathscr F)\cdot H$ are studied and classified.