Recent results on the Kobayashi and Green-Griffiths-Lang conjectures

Contribution to the 16th Takagi lectures in celebration of the 100th anniversary of K.Kodaira's birthThe study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory -- especially through the concepts of curvature and positivity which are central themes in Kodaira's contributions to mathematics. The aim of these lectures is to present recent results concerning 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 $Y$ of $X$ containing all non constant entire curves $f:{\mathbb C}\to X$. Using the formalism of directed varieties and jet bundles, we 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 possible to exploit similar techniques to investigate a famous conjecture of Shoshichi Kobayashi (1970), according to which a generic algebraic hypersurface of dimension $n$ and of sufficiently large degree $d\ge d_n$ in the complex projective space $\bP^{n+1}$ is hyperbolic: in the early 2000's, Yum-Tong Siu proposed a strategy that led in 2015 to a proof based on a clever use of slanted vector fields on jet spaces, combined with Nevanlinna theory arguments. In 2016, the conjecture has been settled in a different way by Damian Brotbek, making a more direct use of Wronskian differential operators and associated multiplier ideals; shortly afterwards, Ya Deng showed how the proof could be modified to yield an explicit value of $d_n$. We give here a short proof based on a drastic simplification of their ideas, along with a further improvement of Deng's bound, namely $d_n=\lfloor\frac{1}{5}(en)^{2n+2}\rfloor$