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 $f^{\ast }\mathcal{O}(1)=\mathcal{O}(q)$.

16 pages, UPDATED a new theorem: Theorem~1.4