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# Endomorphisms of hypersurfaces and other manifolds

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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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##### Manifold

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Endomorphisms of hypersurfaces and other manifolds
Arnaud BEAUVILLE
We prove in this note the following result: Theorem.A smooth complex projective hypersurface of dimension2and de-gree3admits 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 manifoldX whichadmits an endomorphismf: Xis generically ﬁnite (or equivalently surjective),X which of degree>1XfI.vetirm(oprisecoj,)relerenegorahK¨lyalfis actuallyﬁnite: otherwise it contracts some curveC toa point, so that the class of[C] in H (X,Q) is mapped to0 byf; this contradicts the following remark: ∗ −1Lemma1.Letd= degf. The endomorphismsfandd fofH (X,Q)are inverse of each other. This follows from the formulaff=dId . The proof of the theorem stated in the introduction is based on the following result, which appears essentially in [ARV]: N Proposition 1.LetXbe a submanifold ofP, of dimensionn, and letf be an endomorphism ofXsuch thatfOX(1) =OX(m)for some integerm2. Then 1n Ω 2deg(X). cn(X(2))Let us sketch the proof following [ARV]. We ﬁrst observe that the sheaf 1 1 ΩNspanned by its global sections; thereforeΩ (2)(2) is, which is a quotient PX 1 of ΩNis also spanned by its global sections. Let(2) ,σbe a general section of P|X 10 1 Ω (2); thenσand its pull-backf σH (X,Ω (2m)) haveisolated zeroes [ARV, X X lemma 1.1]. Counting these zeroes gives 1 1 (f)ccn(Ω (2m))degnX(2)). X 1