On the logarithmic Kobayashi conjecture

A complex (not necessarily compact) manifold $X$ is said (Brody) hyperbolic if there exists no non-constant entire curves $f:\mathbb{C}\rightarrow X$. In the 70s, Kobayashi conjectured that a general hypersurface $X$ of high degree in the projective space $\mathbb{P}^n$ is hyperbolic. Moreover, he further conjectured that the complement $\mathbb{P}^n\setminus X$ should also be hyperbolic. The first conjecture was proved by Yum-Tong Siu in 2015 and Damian Brotbek in 2016 independently. In this talk, I will present the proof of the second conjecture. The techniques are based on the construction of logarithmic jet differentials, which are the obstructions of the entire curves. The talk is based on joint work with Brotbek.