Generalized polarized manifolds with low second class

Abstract

On a smooth complex projective variety $X$ of dimension $n$, consider an ample vector bundle $\mathcal{E}$ of rank $r\le n-2$ 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 $g:=1+\frac{1}{2}(K_X+c_1(\mathcal{F})+H)\cdot c_{n-2}(\mathcal{F})\cdot H$ are studied and classified.

23 pages; the final presentation of the main topics has been improved with some changes, especially in Table 1, and some typos have been corrected