Galois self-covers of projective spaces and essential dimensions

Abstract

We give the structure theorem of Galois self-covers $f: \mathbf{P}^n \to \mathbf{P}^n$. As an application, we show that the essential dimension of every such nontrivial cover attains its maximum possible value $n$. As another application, we prove that the pair $(\mathbf{P}^n, R_f/(q-1))$ is log Calabi-Yau as conjectured by Gongyo, where $R_f$ is the ramification divisor and we write $f^*\mathcal{O}(1) = \mathcal{O}(q)$.