# 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 ei(PX(E),H(E))for i ≥ 2_n* - 3 or

*i*= 1, and the

*i_th sectional Hodge numberof type (*- 1 and 0 ≤

*j*,*i*-*j*)*hi**i*-*j*(**P***X*(*E*),*H*(*E*)) for*i*≥ 2_n*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*)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