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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
Laboratoire de Mathématiques d'Orsay (LMO) ; Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
International audience
ISSN: 0251-4184
EISSN: 2315-4144
Acta Mathematica Vietnamica
Springer Singapore
hal-03286296
https://universite-paris-saclay.hal.science/hal-03286296
https://universite-paris-saclay.hal.science/hal-03286296/document
https://universite-paris-saclay.hal.science/hal-03286296/file/58.pdf
https://universite-paris-saclay.hal.science/hal-03286296
Acta Mathematica Vietnamica, In press
en
[MATH]Mathematics [math]
info:eu-repo/semantics/article
Journal articles
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
info:eu-repo/semantics/OpenAccess