Endomorphisms of hypersurfaces and other manifolds
By A. Beauville. 2000
71 citations recorded by the source corpus.
We prove in this note the following result: Theorem .− A smooth complex projective hypersurface of dimension ≥ 2 and degree ≥ 3 admits no endomorphism of degree > 1 . Since the case of quadrics is treated in [PS], this settles the question of endomorphisms 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 .
Subfields: Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/d917245181a55f4e5794c2d4acd5f26e3afecf92
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In Geometry and Topology
- De-Qi Zhang
- T. Dinh
- N. Sibony
- Noboru Nakayama
- Shou-wu Zhang
- Tuyen T. T. Truong
- Franccois Berteloot
- Romain Dujardin
Citations in the corpus
- Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-like Mappings (2008, background): “In higher dimension, Beauville proved that a smooth hypersurface of P, k ≥ 3, of degree > 1 does not have an endomorphism with dt > 1, unless the degree is 2, k = 3 and the hypersurface is isomorphic to P(1) × P(1) [7].”
- Distributions in algebraic dynamics (2005, background): “For example, Beauville [6] showed that any smooth hypersurface in projective space with dimension ≥ 2 and degree ≥ 3 does not admit any endomorphism of degree ≥ 2.”
- Polarized endomorphisms of complex normal varieties (2009, background): “Then the following results are known: • d ≤ 2 by [4].”