AUTOMORPHISMS AND MODULI SPACES OF VARIETIES WITH AMPLE CANONICAL CLASS VIA DEFORMATIONS OF ABELIAN COVERS

By a recent result of Viehweg, projective manifolds with ample canonic al class have a coarse moduli space, which is a union of quasiprojecti ve varieties. In this paper, we prove that there are manifolds with am ple canonical class that lie on arbitrarily many irreducible component s of the moduli; moreover, for any finite abelian group G' there exist infinitely many components M of the moduli of varieties with ample ca nonical class such that the generic automorphism group G(M) is equal t o G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension greater than or eq ual to 2. A Galois cover f : X --> Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundl es on Y, satisfying appropriate compatibility conditions. Natural defo rmations of an abelian cover are also introduced in [Pa1]. In this pap er we prove two results about abelian covers: first, that if the build ing data are sufficiently ample, then the natural deformations subject on the Kuranishi family of X; second, that if the building data are s ufficiently ample and generic, then Aut(X) = G. These results, althoug h in some sense ''expected'', are in fact rather powerful and enable u s to construct the required examples. Finally, note that it is essenti al for our applications to be able to deal with general abelian covers and not only with cyclic ones.