Endomorphisms of projective bundles over a certain class of varieties

Ekaterina Amerik; Alexandra Kuznetsova

doi:10.4134/BKMS.b160741

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2017; 54(5): 1743-1755

Online first article July 26, 2017 Printed September 30, 2017

https://doi.org/10.4134/BKMS.b160741

Endomorphisms of projective bundles over a certain class of varieties

Ekaterina Amerik, Alexandra Kuznetsova

Laboratory of Algebraic Geometry and Applications, Laboratory of Algebraic Geometry and Applications

Abstract

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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 $X=\p(E)$. 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.

Keywords: endomophisms, projective bundles, Newton polyhedra

MSC numbers: 14J60, 14L30