Singularities of varieties admitting an endomorphism

  title={Singularities of varieties admitting an endomorphism},
  author={Ama{\"e}l Broustet and Andreas H{\"o}ring},
  journal={Mathematische Annalen},
Let $$X$$X be a normal variety such that $$K_X$$KX is $$\mathbb {Q}$$Q-Cartier, and let $$f:X \rightarrow X$$f:X→X be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that are not log-canonical and the dynamics of the endomorphism $$f$$f. As a consequence we prove that if $$X$$X is projective and $$f$$f polarised, then $$X$$X has at most log-canonical singularities. 
Characterizations of toric varieties via polarized endomorphisms
Let X be a normal projective variety and $$f:X\rightarrow X$$f:X→X a non-isomorphic polarized endomorphism. We give two characterizations for X to be a toric variety. First we show that if X is
Building blocks of amplified endomorphisms of normal projective varieties
  • Sheng Meng
  • Mathematics
    Mathematische Zeitschrift
  • 2019
Let X be a normal projective variety. A surjective endomorphism $$f{:}X\rightarrow X$$ f : X → X is int-amplified if $$f^*L - L =H$$ f ∗ L - L = H for some ample Cartier divisors L and H . This is a
On endomorphisms of projective varieties with numerically trivial canonical divisors
Let $X$ be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism $f:X\to X$ is amplified (resp.~quasi-amplified) if $f^*D-D$ is ample (resp.~big) for some
Surjective endomorphisms of projective surfaces -- the existence of infinitely many dense orbits
Let $f \colon X \to X$ be a surjective endomorphism of a normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we
Totally Invariant Divisors of Int-Amplified Endomorphisms of Normal Projective Varieties
  • Guolei Zhong
  • Mathematics
    The Journal of Geometric Analysis
  • 2020
We consider an arbitrary int-amplified surjective endomorphism f of a normal projective variety X over $$\mathbb {C}$$ C and its $$f^{-1}$$ f - 1 -stable prime divisors. We extend the early result in
Non-isomorphic endomorphisms of Fano threefolds
Let $X$ be a smooth Fano threefold. We show that $X$ admits a non-isomorphic surjective endomorphism if and only if $X$ is either a toric variety or a product of $\mathbb{P}^1$ and a del Pezzo
Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism
Let $X$ be a projective variety admitting a polarized (or more generally, int-amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is
Singularities of non-$\mathbb{Q}$-Gorenstein varieties admitting a polarized endomorphism
In this paper, we discuss a generalization of log canonical singularities in the non-Q-Gorenstein setting. We prove that if a normal complex projective variety has a non-invertible polarized


Endomorphisms of smooth projective $3$-folds with nonnegative Kodaira dimension, II
Let X be a nonsingular projective 3-fold with non-negative Kodaira dimension κ(X) ≥ 0 which admits a nonisomorphic surjective morphism f : X → X onto itself. If κ(X) = 0 or 2, a suitable finite étale
Galois coverings and endomorphisms of projective varieties
We prove that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This results is applied to the study of ramified endomorphisms
On endomorphisms of projective bundles
Abstract. 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
The volume of an isolated singularity
We introduce a notion of volume of a normal isolated singularity that gener- alizes Wahl's characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity
Endomorphisms of hypersurfaces and other manifolds
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
Birational Geometry of Algebraic Varieties
Needless to say, tlie prototype of classification theory of varieties is tlie classical classification theory of algebraic surfaces by the Italian school, enriched by Zariski, Kodaira and others. Let
Invariant hypersurfaces of endomorphisms of the projective 3-space
We consider surjective endomorphisms f of degree > 1 on the projective n-space with n = 3, and f^{-1}-stable hypersurfaces V. We show that V is a hyperplane (i.e., deg(V) = 1) but with four possible
A characteristic number for links of surface singularities
Milnor and Thurston [MT] define a characteristic number of a closed orientable 3-manifold M to be a real-valued topological invariant (r(M) such that: if (p(M) is defined, and M is a k-sheeted
Polarized endomorphisms of uniruled varieties. With an appendix by Y. Fujimoto and N. Nakayama
Abstract We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on ℚ-Fano threefolds, Gorenstein log del
Holomorphic self-maps of singular rational surfaces
We give a new proof of the classification of normal singular surface germs admitting non-invertible holomorphic self-maps and due to J. Wahl. We then draw an analogy between the birational