# On the hyperbolicity of general hypersurfaces

@article{Brotbek2016OnTH,
title={On the hyperbolicity of general hypersurfaces},
author={Damian Brotbek},
journal={Publications math{\'e}matiques de l'IH{\'E}S},
year={2016},
volume={126},
pages={1-34}
}
• Damian Brotbek
• Published 1 April 2016
• Mathematics
• Publications mathématiques de l'IHÉS
In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in Pn$\mathbf {P}^{n}$ are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen such that the geometry of their higher order jet spaces can be related to the geometry of a…
Kobayashi hyperbolicity of the complements of general hypersurfaces of high degree
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• 2019
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• 2018
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• 2020
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## References

SHOWING 1-10 OF 69 REFERENCES
Hyperbolicity of generic high-degree hypersurfaces in complex projective space
We use two ingredients to prove the hyperbolicity of generic hypersurfaces of sufficiently high degree and of their complements in the complex projective space. One is the pullbacks of appropriate
Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces
The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over ${\mathbb C}$, there exists a proper algebraic subvariety of $X$ containing all non constant
Existence of global invariant jet differentials on projective hypersurfaces of high degree
Let $${X\subset\mathbb P^{n+1}}$$ be a smooth complex projective hypersurface. In this paper we show that, if the degree of X is large enough, then there exist global sections of the bundle of
Differential equations on complex projective hypersurfaces of low dimension
Abstract Let n=2,3,4,5 and let X be a smooth complex projective hypersurface of $\mathbb {P}^{n+1}$. In this paper we find an effective lower bound for the degree of X, such that every holomorphic
Hyperbolicity in Complex Geometry
A complex manifold is said to be hyperbolic if there exists no nonconstant holomorphic map from the affine complex line to it. We discuss the techniques and methods for the hyperbolicity problems for
Hyperbolicity related problems for complete intersection varieties
Abstract In this paper we examine different problems regarding complete intersection varieties of high multidegree in a smooth complex projective variety. First we prove an existence theorem for jet
Algebraic families of smooth hyperbolic surfaces of low degree in ℙℂ3
We reprove (after a paper of Y.T. Siu appeared in 1987) a simple vanishing theorem for the Wronskian of Brody curves under a suitable assumption on the existence of global meromorphic connections.
Subvarieties of general type on a general projective hypersurface
We study subvarieties of a general projective degree $d$ hypersurface $X_d\subset \mathbf P^n$. Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the
Compact manifolds in hyperbolicity
In this paper we establish the strongest possible criterion for the hyperbolicity of a compact complex manifold: such a manifold is hyperbolic if and only if it contains no (nontrivial) complex
A remark on the codimension of the Green–Griffiths locus of generic projective hypersurfaces of high degree
• Mathematics
• 2009
Abstract We show that for every smooth generic projective hypersurface , there exists a proper subvariety such that codim X Y ≧ 2 and for every non constant holomorphic entire map ƒ : ℂ → X one has ƒ