Kobayashi hyperbolicity in degree > n^{2n}

TL;DR

This paper proves that generic hypersurfaces of degree greater than n^{2n} in complex projective space are Kobayashi hyperbolic, improving previous bounds and establishing hyperbolicity of their complements.

Contribution

It establishes new lower bounds on the degree for hyperbolicity of hypersurfaces and their complements, surpassing prior results by Demailly and Brotbek-Deng.

Findings

Hypersurfaces of degree > n^{2n} are Kobayashi hyperbolic.

Complements of these hypersurfaces are Kobayashi hyperbolically embedded.

The bounds are improved over previous known results.

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 ${\sf const} > 1$.