Totally invariant divisors of endomorphisms of projective spaces

@article{Hring2016TotallyID,
  title={Totally invariant divisors of endomorphisms of projective spaces},
  author={Andreas H{\"o}ring},
  journal={manuscripta mathematica},
  year={2016},
  volume={153},
  pages={173-182}
}
  • A. Höring
  • Published 24 January 2016
  • Mathematics
  • manuscripta mathematica
Totally invariant divisors of endomorphisms of the projective space are expected to be always unions of linear spaces. Using logarithmic differentials we establish a lower bound for the degree of the non-normal locus of a totally invariant divisor. As a consequence we prove the linearity of totally invariant divisors for $$\mathbb {P}^3$$P3. 
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References

SHOWING 1-10 OF 14 REFERENCES
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
On Endomorphisms of Fano Manifolds of Picard Number One
Let X be a Fano manifold of Picard number one different from the projective space. It has been conjectured that a surjective endomorphism X → X must be bijective. In this article, we will prove a
Logarithmic sheaves attached to arrangements of hyperplanes
A reduced divisor on a nonsingular variety defines the sheaf of logarithmic 1-forms. We introduce a certain coherent sheaf whose double dual coincides with this sheaf. It has some nice properties,
Polarized endomorphisms of complex normal varieties
It is shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism. The geometric structure of the variety is described
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
Linearity of the exceptional set for maps of Pk(C)
Abstract.Let f be a holomorphic self-map of Pk(C), with degree larger than 2. We show that its exceptional set is a finite union of linear subspaces.
Singularities of varieties admitting an endomorphism
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
Hypersurfaces exceptionnelles des endomorphismes de ℂℙ(n)
ResuméOn étudie les hypersurfaces exceptionnelles pour les applications holomorphes de ℂℙ(n). On montre qu'une telle hypersurface n'est jamais lisse dès que son degré est plus grand que
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