Abstract:
Let $B$ be a simply-connected projective variety such that the first
cohomology groups of all line bundles on $B$ are zero. Let $E$ be a vector
bundle over $B$ and $X={\mathbb P} (E)$. It is easily seen that a power of
any endomorphism of $X$ takes fibers to fibers. We prove that if $X$ admits
an endomorphism which is of degree greater than one on the fibers then $E$
splits into a direct sum of line bundles.
