On the Structure of Generalized Polarized Manifolds with Relatively Small Second Class

Abstract

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 $\mathcal{E}$ of rank $r\leq n-2$ on $X$ such that $\mathcal{F}:=\mathcal{E}\oplus H^{\oplus (n-r-2)}$ 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 $m_2=m_2(X,\mathcal{E},H)$ is the generalized second class defined in a previous paper and $d:=c_{n-2}(\mathcal{F})H^2$.