Kobayashi hyperbolicity in degree > n²ⁿ

2018/07/30 by Merker, Joël
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.11309

Abstract

For a generic hypersurface \mathbbX^n-1 ⊂ ℙⁿ(ℂ) of degree d \geqslant n²ⁿ (1) ℙⁿ \backslash \mathbbX^n-1 is Kobayashi-hyperbolically imbedded in ℙⁿ; (2) \mathbbX^n-1 is Kobayashi(⇔ Brody)-hyperbolic. (1) improves Brotbek-Deng 1804.01719: d \geqslant (n+2)ⁿ⁺³ (n+1)ⁿ⁺³ = n²ⁿ n⁶ (e³+\rm O((1)/(n)) ). (2) supersedes Demailly 1801.04765: d \geqslant (1)/(3) ( e¹(n-1) )²ⁿ = n²ⁿ e²ⁿ ( (1)/(3 e²) + \rm O ((1)/(n)) ). The method gives in fact d \geqslant \fracn²ⁿ\sf constⁿ for n \geqslant N(\sf const) with any \sf const > 1.

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