The 16th Takagi Lectures
 November 28, 2015 (Sat) 15:45--16:45 November 29, 2015 (Sun) 14:00--15:00 Graduate School of Mathematical Sciences The University of Tokyo, Tokyo, Japan

Recent Progress Towards the Kobayashi
and Green--Griffiths--Lang Conjectures
Jean-Pierre Demailly
(Université de Grenoble I)

Abstract

The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory. The aim of the lectures is to present recent progress on the geometric side of the problem.

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 will show 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 $T_X$. It is then possible to exploit this result to investigate the long-standing conjecture of Kobayashi (1970), according to which every general algebraic hypersurface of dimension $n$ and degree at least $2n + 1$ in the complex projective space ${\mathbb P}^{n+1}$ is hyperbolic.