  We prove a connexity theorem for abelian varieties in characteristic $0$: if
$X$ is an abelian variety and $V\rightarrow X$ and $W\rightarrow X$ two
morphisms, then, under certain hypotheses, the fiber product of $V$ and $W$
over $X$ is connected. This theorem has several consequences: the algebraic
fundamental groups of a simple abelian variety $X$ and of a normal subvariety
$V$ of $X$ of dimension $> dim(X)/2$ are isomorphic. The same holds when $V$ is
a finite cover of degree $\le dim(X)$ of $X$. (The only difference between this
replacement and the original submission is that one of the conjectures is now
proved).
