On endomorphisms of projective bundles

Abstract

Let $X$ be a complex projective bundle. We prove that $X$ admits an endomorphism of degree $>1$ and commuting with the projection to the base, if and only if $X$ trivializes after a finite covering. When $X$ is the projectivization of a vector bundle $E$ of rank 2, we prove that it has an endomorphism of degree $>1$ on a general fiber only if $E$ splits after a finite base change.