On Abelian Canonical $n$-folds of General Type

Abstract

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 $n \geqslant 4$ 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 $K_X$ defines an abelian cover over $\mathbb{P}^n$, i.e., when $X$ is an \emph{abelian canonical $n$-fold}, then the canonical degree of $X$ is universally upper bounded by a constant which only depends on $n$ for $X$ non-singular. We also construct two examples of non-singular minimal projective $4$-folds of general type with canonical degrees $81$ and $128$.