Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces

2015/01/28 by Demailly, Jean-Pierre
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1501.07625

Abstract

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→ 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 Pⁿ⁺¹ is hyperbolic.

Related

• Recent results on the Kobayashi and Green-Griffiths-Lang conjectures
• Towards the Green-Griffiths-Lang conjecture
• On the hyperbolicity of general hypersurfaces
• A remark on the codimension of the Green-Griffiths locus of generic projective hypersurfaces of high degree
• Nondeformability of entire curves in projective hypersurfaces of high degree