# Sectional Invariants of Scroll over a Smooth Projective Variety

### Yoshiaki Fukuma

Kochi University, Japan

## Abstract

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

## Cite this article

Yoshiaki Fukuma, Sectional Invariants of Scroll over a Smooth Projective Variety. Rend. Sem. Mat. Univ. Padova 121 (2009), pp. 93–119

DOI 10.4171/RSMUP/121-6