Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces
- Resource Type
- Working Paper
- Authors
- Demailly, Jean-Pierre
- Source
- Subject
- Mathematics - Algebraic Geometry
Mathematics - Complex Variables
- Language
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 entire curves $f:{\mathbb C}\to X$. 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 $T\_X$. We then use this fact to confirm a long-standing conjecture of Kobayashi (1970), according to which a very general algebraic hypersurface of dimension $n$ and degree at least $2n+2$ in the complex projective space ${\mathbb P}^{n+1}$ is hyperbolic.
Comment: This paper supersedes submission hal-01092537 / arXiv:1412.2986