Sectional Invariants of Scroll over a Smooth Projective Variety

Abstract

Let $X$ be a smooth complex variety of dimension $n$ and let $\mathcal{E}$ be an ample vector bundle of rank $r$ on $X$. Then we calculate the $i$th sectional Euler number $e_i({\mathbb{P}}_X(\mathcal{E}),H(\mathcal{E}))$ for $i\ge 2n-3$ or $i=1$, and the $i$th sectional Hodge number of type $(j,i-j)h^{j,i-j}({\mathbb{P}}_X(\mathcal{E}),H(\mathcal{E}))$ for $i\ge 2n-1$ and $0\le j\le i$, where ${\mathbb{P}}_X(\mathcal{E})$ is the projective space bundle associated with $\mathcal{E}$ and $H(\mathcal{E})$ is its tautological line bundle. Moreover we define a new invariant $v(X,\mathcal{E})$ of $(X,\mathcal{E})$ for $r\ge n-1$. This invariant is thought to be a generalization of curve genus. We will investigate some properties of this invariant.