# ENDOMORPHISMS OF PROJECTIVE BUNDLES OVER A CERTAIN CLASS OF VARIETIES

• Amerik, Ekaterina (National Research University Higher School of Economics Laboratory of Algebraic Geometry and Applications) ;
• Kuznetsova, Alexandra (National Research University Higher School of Economics Laboratory of Algebraic Geometry and Applications)
• Accepted : 2017.04.05
• Published : 2017.09.30

#### 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.

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