On abelian canonical n-folds of general type

Let X be a Gorenstein minimal projective n-fold with at worst locally factorial terminal singularities, and suppose that the canonical map of X is generically finite onto its image. When n < 4, the canonical degree is universally bounded. While the possibility of obtaining a universal bound on the canonical degree of X for n ≥ 4 may be inaccessible, we give a uniform upper bound for the degrees of certain abelian covers. In particular, we show that if the canonical divisor KX defines an abelian cover over ℙⁿ, i.e., when X is an abelian canonical n-fold, then the canonical degree of X is universally upper bounded by a constant which only depends on n for X nonsingular. We also construct two examples of nonsingular minimal projective 4-folds of general type with canonical degrees 81 and 128.