Log Calabi--Yau structure for endomorphisms on $\mathbf{P}^n$

Abstract

Let $f:\mathbf{P}^n\to\mathbf{P}^n$ be a $q$-polarized endomorphism, where $q>1$, and let $R_f$ be its ramification divisor. We study the singularities of the ramification pair $(\mathbf{P}^n,R_f)$. We show that, for a general $f$, the pair $(\mathbf{P}^n,R_f)$ is log canonical. When $n=2$, we prove that there exists an integer $s\geq1$ such that the log canonical threshold $\mathrm{lct}(\mathbf{P}^2;R_{f^s})\geq1/(q^s-1)$. The passage to an iterate is necessary in general, and the lower bound is optimal. In particular, $(\mathbf{P}^2,R_{f^s}/(q^s-1))$ is a log Calabi--Yau pair, completing the proof of Gongyo's conjecture for smooth projective surfaces.