%0 Journal Article
%T Degrees d > (sqrt(n) log n)^n and d > (n log n)^n in the Conjectures of Green-Griffiths and of Kobayashi
%+ Laboratoire de Mathématiques d'Orsay (LMO)
%A Merker, Joël
%A Ta, The-Anh
%< avec comité de lecture
%@ 0251-4184
%J Acta Mathematica Vietnamica
%I Springer Singapore
%8 2021
%D 2021
%Z Mathematics [math]Journal articles
%X Once first answers in any dimension to the Green-Griffiths and Kobayashi conjecturesfor generic algebraic hypersurfaces X^n−1 in P^n(C) have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the "celestial" horizon lies near d > 2n.For Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain:d > (sqrt(n) log n)^n,and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain:d > (n log n)^n.The latter improves d > n^2n obtained by Merker in arxiv.org/1807/11309/.Admitting a certain technical conjecture I0 > I0', the method employed (Diverio-Merker-Rousseau, Bérczi, Darondeau) conducts to constant power n, namely to:d > 2^5nand, respectively, to:d > 4^5n.In Spring 2021, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that I0 > I0', a conjecture which will be established up to dimension n = 50.
%G English
%2 https://universite-paris-saclay.hal.science/hal-03286296/document
%2 https://universite-paris-saclay.hal.science/hal-03286296/file/58.pdf
%L hal-03286296
%U https://universite-paris-saclay.hal.science/hal-03286296
%~ CNRS
%~ INSMI
%~ LM-ORSAY
%~ UNIV-PARIS-SACLAY
%~ TEST-HALCNRS
%~ UNIVERSITE-PARIS-SACLAY
%~ GS-MATHEMATIQUES
%~ TEST3-HALCNRS