Kobayashi hyperbolicity in degree > n^{2n}

Abstract

For a generic hypersurface $\mathbb{X}^{n-1} \subset \mathbb{P}^n(\mathbb{C})$ of degree \[ d \,\geqslant\, n^{2n} \] (1) $\mathbb{P}^n \big\backslash \mathbb{X}^{n-1}$ is Kobayashi-hyperbolically imbedded in $\mathbb{P}^n$; (2) $\mathbb{X}^{n-1}$ is Kobayashi($\Leftrightarrow$ Brody)-hyperbolic. (1) improves Brotbek-Deng 1804.01719: $d \geqslant (n+2)^{n+3}\, (n+1)^{n+3} = n^{2n}\,n^6\, \big(e^3+{\rm O}(\frac{1}{n}) \big)$. (2) supersedes Demailly 1801.04765: $d \geqslant \frac{1}{3}\, \big( e^1(n-1) \big)^{2n} = n^{2n}\, e^{2n}\, \big( \frac{1}{3\, e^2} + {\rm O} (\frac{1}{n}) \big)$. The method gives in fact $d \geqslant \frac{n^{2n}}{{\sf const}^n}$ for $n \geqslant N({\sf const})$ with any ${\sf const} > 1$.