Generalized polarized manifolds with low second class

TL;DR

This paper introduces a new numerical invariant for polarized manifolds with ample vector bundles, extending the second class concept, and classifies cases with low invariant values under certain conditions.

Contribution

It defines a generalized second class invariant for polarized manifolds with ample vector bundles and classifies low-invariant cases under specific assumptions.

Findings

Classification of triplets with small second class invariant

Extension of second class concept to higher dimensions and vector bundles

Conditions under which the classification applies

Abstract

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.