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[CNRS] CNRS - Centre national de la recherche scientifique
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[LM-ORSAY] Laboratoire de Mathématiques d'Orsay
[UNIV-PARIS-SACLAY] Université Paris-Saclay
[TEST-HALCNRS] Collection test HAL CNRS
[UNIVERSITE-PARIS-SACLAY] Université Paris-Saclay
[GS-MATHEMATIQUES] Graduate School Mathématiques
[TEST3-HALCNRS] TEST3-HALCNRS
Degrees d > (sqrt(n) log n)^n and d > (n log n)^n in the Conjectures of Green-Griffiths and of Kobayashi
Merker, Joël
Ta, The-Anh
[MATH] Mathematics [math]
ART
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.
2021
2021-07-14
en
Acta Mathematica Vietnamica
Springer Singapore