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 |
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.