https://universite-paris-saclay.hal.science/hal-03286296Merker, JoëlJoëlMerkerLMO - Laboratoire de Mathématiques d'Orsay - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueTa, The-AnhThe-AnhTaLMO - Laboratoire de Mathématiques d'Orsay - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueDegrees d > (sqrt(n) log n)^n and d > (n log n)^n in the Conjectures of Green-Griffiths and of KobayashiHAL CCSD2021[MATH] Mathematics [math]MERKER, Joël2021-07-14 10:28:352023-03-24 14:53:222021-07-19 08:54:14enJournal articlesapplication/pdf1Once 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.