Totally invariant subvarieties of $\mathbf P^n$

Abstract

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.