  By a recent result of Viehweg, projective manifolds with ample canonical
class have a coarse moduli space, which is a union of quasiprojective
varieties. In this paper, we prove that there are manifolds with ample
canonical class that lie on arbitrarily many irreducible components of the
moduli; moreover, for any finite abelian group $G$ there exist infinitely many
components $M$ of the moduli of varieties with ample canonical class such that
the generic automorphism group $G_M$ is equal to $G$. In order to construct the
examples, we use abelian covers, i.e. Galois cover whose Galois group is finite
and abelian. We prove two results about abelian covers: first, that if the
building data are sufficiently ample, then the natural deformations surject on
the Kuranishi family of $X$; second, that if the building data are sufficiently
ample and generic, then $Aut(X)=G$.
