Towards the Green-Griffiths-Lang conjecture

@article{Demailly2014TowardsTG,
  title={Towards the Green-Griffiths-Lang conjecture},
  author={J Demailly},
  journal={arXiv: Algebraic Geometry},
  year={2014}
}
  • J. Demailly
  • Published 9 December 2014
  • Mathematics
  • arXiv: Algebraic Geometry
The Green-Griffiths-Lang conjecture stipulates that for every projective variety X of general type over C, there exists a proper algebraic subvariety of X containing all non constant entire curves f : C $\rightarrow$ X. Using the formalism of directed varieties, we prove here that this assertion holds true in case X satisfies a strong general type condition that is related to a certain jet-semistability property of the tangent bundle TX . We then give a sufficient criterion for the Kobayashi… 
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Methods Citations
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Recent results on the Kobayashi and Green-Griffiths-Lang conjectures
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  • Mathematics
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References

SHOWING 1-10 OF 18 REFERENCES
Holomorphic Morse Inequalities and the Green-Griffiths-Lang Conjecture
The goal of this work is to study the existence and properties of non constant entire curves f drawn in a complex irreducible n-dimensional variety X, and more specifically to show that they must…
The exceptional set and the Green-Griffiths locus do not always coincide
We give a very simple criterion in order to ensure that the Green-Griffiths locus of a projective manifold is the whole manifold. Next, we use it to show that the Green-Griffiths locus of any…
Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials
These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995. The main goal of the notes is to study complex varieties…
Some recent transcendental techniques in algebraic and complex geometry
This article discusses the recent transcendental techniques used in the proofs of the following three conjectures. (1)~The plurigenera of a compact projective algebraic manifold are invariant under…
Diophantine approximations and foliations
In this paper we indicate the proof of an effective version of the Green-Griffiths conjecture for surfaces of general type and positive second Segre class (i.e.c12>c2). Naturally this effective…
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 of generic surfaces of high degree in projective 3-space
<abstract abstract-type="TeX"><p>The main goal of this work is to prove that a very generic surface of degree at least 21 in complex projective 3-dimensional space is hyperbolic in the sense of…
Holomorphic Curves on Hyperplane Sections of 3-Folds
Abstract. In this paper we prove a conjectured height inequality of Lang and Vojta for holomorphic curves lying on generic hyperplane sections of 3-folds. As a consequence we deduce a conjecture of…
Effective algebraic degeneracy
AbstractWe show that for every smooth projective hypersurface X⊂ℙn+1 of degree d and of arbitrary dimension n≥2, if X is generic, then there exists a proper algebraic subvariety Y⊊X such that every…
Hyperbolic complex spaces
In the three decades since the introduction of the Kobayashi distance, the subject of hyperbolic complex spaces and holomorphic mappings has grown to be a big industry. This book gives a…
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