Endomorphisms of projective bundles over a certain class of varieties
Ekaterina Amerik, Alexandra Kuznetsova

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 be a simply-connected projective variety such that the first cohomology groups of all line bundles on are zero. Let be a vector bundle over and . It is easily seen that a power of any endomorphism of takes fibers to fibers. We prove that if admits an endomorphism which is of degree greater than one on the fibers then splits into a direct sum of line bundles.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Geometry and complex manifolds
