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Let $f:\mathbf P^n_{\mathbb C}\to\mathbf P^n_{\mathbb C}$ be an endomorphism of degree greater than one. We prove that every irreducible Zariski closed subset $V$ with $f^{-1}(V)=V$ is a linear subspace.
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 a...
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