Degrees d > (sqrt(n) log n)^n and d > (n log n)^n in the Conjectures of Green-Griffiths and of Kobayashi

Joël Merker (1) , The-Anh Ta (1)

1 LMO - Laboratoire de Mathématiques d'Orsay

Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for 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 > 2^5n and, 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.

Domaines: Mathématiques [math]

Joël Merker, The-Anh Ta. Degrees d > (sqrt(n) log n)^n and d > (n log n)^n in the Conjectures of Green-Griffiths and of Kobayashi. Acta Mathematica Vietnamica, In press. ⟨hal-03286296⟩