On endomorphisms of projective bundles

Let X be a projective bundle. We prove that X admits an endomorphism of degree > 1 and commuting with the projection to the base, if and only if X trivializes after a finite covering. When X is the projectivization of a vector bundle E of rank 2, we prove that it has an endomorphism of degree > 1 on a general fiber only if E splits after a finite base change. It is clear that, for a complex projective variety X, the existence of endomorphisms f : X → X of degree bigger than one imposes very strong restrictions on the geometry of X. On the other hand, for any variety B, the product B × P has such endomorphisms. So it might be interesting to study the following question: Let X be a projective bundle over a smooth projective complex variety B, p : X → B the projection map. When does X admit a surjective endomorphism of degree bigger than one? This question is the subject of the present article. Remark that any surjective endomorphism f of X is finite. Indeed, the inverse image map f ∗ on the rational cohomologies of X is injective, because f∗f ∗ = deg(f) · id, so an isomorphism, but if f contracts a curve, this map obviously cannot be surjective. We will prove the following Theorem 1 X admits an endomorphism over B (i.e. commuting with p) and of degree bigger than one if and only if X trivializes after a finite base change. ∗Université Paris-Sud, Laboratoire des Mathématiques, Bâtiment 425, 91405 Orsay, France. [email protected]