Abelian Covers and Isotrivial Canonical Fibrations

We give a pure algebraic method to construct all the infinite families of surfaces S with isotrivial canonical fibration where S is the minimal desingularization of X = Z/G and G is an Abelian group acting diagonally on the product of two smooth curves: Z = F × D. In particular we recover all the known infinite families of surfaces with isotrivial canonical fibration and we produce many new ones. Our method works in every dimension and, with minor modifications, it can be applied to construct surfaces with canonical map of degree > 1.