5 Pages
English

# Endomorphisms of hypersurfaces and other manifolds

- Description

Endomorphisms of hypersurfaces and other manifolds Arnaud BEAUVILLE We prove in this note the following result: Theorem .? A smooth complex projective hypersurface of dimension ≥ 2 and de- gree ≥ 3 admits no endomorphism of degree > 1 . Since the case of quadrics is treated in [PS], this settles the question of endo- morphisms of hypersurfaces. We prove the theorem in Section 1, using a simple but efficient trick devised by Amerik, Rovinsky and Van de Ven [ARV]. In Section 2 we collect some general results on endomorphisms of projective manifolds; we classify in particular the Del Pezzo surfaces which admit an endomorphism of degree > 1 . I am indebted to I. Dolgachev for bringing the problem to my attention. 1. Hypersurfaces We will consider in this note a compact complex manifold X which admits an endomorphism f : X ? X which is generically finite (or equivalently surjective), of degree > 1 . If X is projective (or more generally Kahler), f is actually finite : otherwise it contracts some curve C to a point, so that the class of [C] in H?(X,Q) is mapped to 0 by f? ; this contradicts the following remark: Lemma 1 .? Let d = deg f . The endomorphisms f? and d?1f? of H ?(X,Q) are inverse of each other.

• onto compact

• ks ?

• dimension

• every kodaira

• projective manifold

• manifold

• gree ≥

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