## Vertical vector fields on certain complex fibrations

J. Math. Sci. Univ. Tokyo
Vol. 11 (2004), No. 3, Page 177--243.

Rebelo, Julio C.
Vertical vector fields on certain complex fibrations
The first part of this paper is devoted to constructing a certain special meromorphic vector field over a given complex fibration. We consider the case of ruled/elliptic fibrations in arbitrary dimensions and the case of hyperelliptic fibrations in dimension~$2$. The second part of the paper uses the existence of these vector fields to classify the singularities of a hyperelliptic fibration and, in particular, of a fibration whose generic fiber has genus~$2$. This classification will be employed in a continuation of this work to yield models for fibrations with genus~$2$. Our construction might also find applications to the study of elliptic threefolds.