Endomorphisms of projective bundles over a certain class of varieties

TL;DR

This paper investigates endomorphisms of projective bundles over certain simply-connected varieties, showing that high-degree fiber endomorphisms imply the vector bundle splits into line bundles.

Contribution

It proves that if a projective bundle admits a fiber-degree greater than one endomorphism, then the underlying vector bundle splits into line bundles, extending understanding of endomorphism structures.

Findings

Endomorphisms of degree > 1 on fibers imply bundle splitting

Fiber-preserving endomorphisms are characterized in the setting

Splitting criterion applies to a class of simply-connected varieties

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.