Kobayashi hyperbolicity in degree > n^{2n}

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($\iff$ 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 $d⩾\frac{n^{2n}}{{\mathsf{c}\mathsf{o}\mathsf{n}\mathsf{s}\mathsf{t}}^n}$ for $n \geqslant N({\sf const})$withany${\sf const} > 1$.